![Page 1: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/1.jpg)
Moonshine: Lecture 3
Moonshine: Lecture 3
Ken Ono (Emory University)
![Page 2: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/2.jpg)
Moonshine: Lecture 3
I’m going to talk about...
I. History of Moonshine
II. Distribution of Monstrous Moonshine
III. Umbral Moonshine
![Page 3: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/3.jpg)
Moonshine: Lecture 3
I’m going to talk about...
I. History of Moonshine
II. Distribution of Monstrous Moonshine
III. Umbral Moonshine
![Page 4: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/4.jpg)
Moonshine: Lecture 3
I’m going to talk about...
I. History of Moonshine
II. Distribution of Monstrous Moonshine
III. Umbral Moonshine
![Page 5: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/5.jpg)
Moonshine: Lecture 3
I’m going to talk about...
I. History of Moonshine
II. Distribution of Monstrous Moonshine
III. Umbral Moonshine
![Page 6: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/6.jpg)
Moonshine: Lecture 3
I. History of Moonshine
The Monster
Conjecture (Fischer and Griess (1973))
There is a huge simple group (containing a double cover ofFischer’s B) with order
246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71.
Theorem (Griess (1982))
The Monster group M exists.
![Page 7: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/7.jpg)
Moonshine: Lecture 3
I. History of Moonshine
The Monster
Conjecture (Fischer and Griess (1973))
There is a huge simple group (containing a double cover ofFischer’s B) with order
246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71.
Theorem (Griess (1982))
The Monster group M exists.
![Page 8: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/8.jpg)
Moonshine: Lecture 3
I. History of Moonshine
The Monster
Conjecture (Fischer and Griess (1973))
There is a huge simple group (containing a double cover ofFischer’s B) with order
246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71.
Theorem (Griess (1982))
The Monster group M exists.
![Page 9: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/9.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Classification of Finite Simple Groups
Theorem (Classification of Finite Simple Groups)
Finite simple groups live in natural infinite families, apart from thesporadic groups.
![Page 10: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/10.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Classification of Finite Simple Groups
Theorem (Classification of Finite Simple Groups)
Finite simple groups live in natural infinite families, apart from thesporadic groups.
![Page 11: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/11.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Classification of Finite Simple Groups
Theorem (Classification of Finite Simple Groups)
Finite simple groups live in natural infinite families, apart from thesporadic groups.
![Page 12: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/12.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Ogg’s Theorem
Theorem (Ogg, 1974)
X0(N) est hyperelliptique pour exactement dix-neuf valuers de N.
Corollary (Ogg, 1974)
Toutes les valuers supersingulieres de j sont Fp si, et seulement si,g+ = 0,
i.e. p ∈ Oggss := {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}.
![Page 13: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/13.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Ogg’s Theorem
Theorem (Ogg, 1974)
X0(N) est hyperelliptique pour exactement dix-neuf valuers de N.
Corollary (Ogg, 1974)
Toutes les valuers supersingulieres de j sont Fp si, et seulement si,g+ = 0,
i.e. p ∈ Oggss := {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}.
![Page 14: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/14.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Ogg’s Theorem
Theorem (Ogg, 1974)
X0(N) est hyperelliptique pour exactement dix-neuf valuers de N.
Corollary (Ogg, 1974)
Toutes les valuers supersingulieres de j sont Fp si, et seulement si,g+ = 0,
i.e. p ∈ Oggss := {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}.
![Page 15: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/15.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Ogg’s Theorem
Theorem (Ogg, 1974)
X0(N) est hyperelliptique pour exactement dix-neuf valuers de N.
Corollary (Ogg, 1974)
Toutes les valuers supersingulieres de j sont Fp si, et seulement si,g+ = 0,
i.e. p ∈ Oggss := {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}.
![Page 16: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/16.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Ogg’s Jack Daniels Problem
Remark
This is the first hint of Moonshine.
![Page 17: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/17.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Ogg’s Jack Daniels Problem
Remark
This is the first hint of Moonshine.
![Page 18: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/18.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Second hint of moonshine
John McKay observed that
196884 = 1 + 196883
![Page 19: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/19.jpg)
Moonshine: Lecture 3
I. History of Moonshine
John Thompson’s generalizations
Thompson further observed:
196884 = 1 + 196883
21493760 = 1 + 196883 + 21296876
864299970 = 1 + 1 + 196883 + 196883 + 21296876 + 842609326864299970︸ ︷︷ ︸
Coefficients of j(τ)
1 + 1 + 196883 + 196883 + 21296876 + 842609326︸ ︷︷ ︸Dimensions of irreducible representations of the Monster M
![Page 20: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/20.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Klein’s j-function
Definition
Klein’s j-function
j(τ)− 744 =∞∑
n=−1
c(n)qn
= q−1 + 196884q + 21493760q2 + 864299970q3 + . . .
satisfies
j
(aτ + b
cτ + d
)= j(τ) for every matrix
(a bc d
)∈ SL2(Z).
![Page 21: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/21.jpg)
Moonshine: Lecture 3
I. History of Moonshine
The Monster characters
The character table for M (ordered by size) gives dimensions:
χ1(e) = 1
χ2(e) = 196883
χ3(e) = 21296876
χ4(e) = 842609326
...
χ194(e) = 258823477531055064045234375.
![Page 22: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/22.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Monster module
Conjecture (Thompson)
There is an infinite-dimensional graded module
V \ =∞⊕
n=−1
V \n
withdim(V \
n) = c(n).
![Page 23: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/23.jpg)
Moonshine: Lecture 3
I. History of Moonshine
The McKay-Thompson Series
Definition (Thompson)
Assuming the conjecture, if g ∈M, then define theMcKay–Thompson series
Tg (τ) :=∞∑
n=−1
tr(g |V \n)qn.
![Page 24: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/24.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Conway and Norton
Conjecture (Monstrous Moonshine)
For each g ∈M there is an explicit genus 0 discrete subgroupΓg ⊂ SL2(R)
for which Tg (τ) is the unique modular function with
Tg (τ) = q−1 + O(q).
![Page 25: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/25.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Conway and Norton
Conjecture (Monstrous Moonshine)
For each g ∈M there is an explicit genus 0 discrete subgroupΓg ⊂ SL2(R) for which Tg (τ) is the unique modular function with
Tg (τ) = q−1 + O(q).
![Page 26: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/26.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Borcherds’ work
Theorem (Frenkel–Lepowsky–Meurman)
The moonshine module V \ =⊕∞
n=−1 V\n is a vertex operator
algebra whose graded dimension is given by j(τ)− 744, and whoseautomorphism group is M.
Theorem (Borcherds)
The Monstrous Moonshine Conjecture is true.
Remark
Earlier work of Atkin, Fong and Smith numerically confirmedMonstrous moonshine.
![Page 27: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/27.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Borcherds’ work
Theorem (Frenkel–Lepowsky–Meurman)
The moonshine module V \ =⊕∞
n=−1 V\n is a vertex operator
algebra whose graded dimension is given by j(τ)− 744, and whoseautomorphism group is M.
Theorem (Borcherds)
The Monstrous Moonshine Conjecture is true.
Remark
Earlier work of Atkin, Fong and Smith numerically confirmedMonstrous moonshine.
![Page 28: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/28.jpg)
Moonshine: Lecture 3
I. History of Moonshine
Borcherds’ work
Theorem (Frenkel–Lepowsky–Meurman)
The moonshine module V \ =⊕∞
n=−1 V\n is a vertex operator
algebra whose graded dimension is given by j(τ)− 744, and whoseautomorphism group is M.
Theorem (Borcherds)
The Monstrous Moonshine Conjecture is true.
Remark
Earlier work of Atkin, Fong and Smith numerically confirmedMonstrous moonshine.
![Page 29: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/29.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question A
Do order p elements in M know the Fp supersingular j-invariants?
Theorem (Dwork’s Generating Function)
If p ≥ 5 is prime, then
(j(τ)− 744) | U(p) ≡
−∑α∈SSp
Ap(α)
j(τ)− α−
∑g(x)∈SS∗
p
Bp(g)j(τ) + Cp(g)
g(j(τ))(mod p).
![Page 30: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/30.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question A
Do order p elements in M know the Fp supersingular j-invariants?
Theorem (Dwork’s Generating Function)
If p ≥ 5 is prime, then
(j(τ)− 744) | U(p) ≡
−∑α∈SSp
Ap(α)
j(τ)− α−
∑g(x)∈SS∗
p
Bp(g)j(τ) + Cp(g)
g(j(τ))(mod p).
![Page 31: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/31.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question A
Do order p elements in M know the Fp supersingular j-invariants?
Theorem (Dwork’s Generating Function)
If p ≥ 5 is prime, then
(j(τ)− 744) | U(p) ≡
−∑α∈SSp
Ap(α)
j(τ)− α−
∑g(x)∈SS∗
p
Bp(g)j(τ) + Cp(g)
g(j(τ))(mod p).
![Page 32: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/32.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Answer to Question A
If g ∈M and p is prime, then Moonshine implies that
Tg + pTg | U(p) = Tgp .
And so if g has order p, then
Tg + pTg | U(p) = j − 744.
Which implies that
Tg ≡ j − 744 (mod p).
....giving us Dwork’s generating function
Tg | U(p) ≡ (j − 744) | U(p) (mod p).
![Page 33: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/33.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Answer to Question A
If g ∈M and p is prime, then Moonshine implies that
Tg + pTg | U(p) = Tgp .
And so if g has order p, then
Tg + pTg | U(p) = j − 744.
Which implies that
Tg ≡ j − 744 (mod p).
....giving us Dwork’s generating function
Tg | U(p) ≡ (j − 744) | U(p) (mod p).
![Page 34: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/34.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Answer to Question A
If g ∈M and p is prime, then Moonshine implies that
Tg + pTg | U(p) = Tgp .
And so if g has order p, then
Tg + pTg | U(p) = j − 744.
Which implies that
Tg ≡ j − 744 (mod p).
....giving us Dwork’s generating function
Tg | U(p) ≡ (j − 744) | U(p) (mod p).
![Page 35: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/35.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Answer to Question A
If g ∈M and p is prime, then Moonshine implies that
Tg + pTg | U(p) = Tgp .
And so if g has order p, then
Tg + pTg | U(p) = j − 744.
Which implies that
Tg ≡ j − 744 (mod p).
....giving us Dwork’s generating function
Tg | U(p) ≡ (j − 744) | U(p) (mod p).
![Page 36: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/36.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Answer to Question A
If g ∈M and p is prime, then Moonshine implies that
Tg + pTg | U(p) = Tgp .
And so if g has order p, then
Tg + pTg | U(p) = j − 744.
Which implies that
Tg ≡ j − 744 (mod p).
....giving us Dwork’s generating function
Tg | U(p) ≡ (j − 744) | U(p) (mod p).
![Page 37: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/37.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question B
If p 6∈ Oggss , then why do we expect p - #M?
Answer
By Moonshine, if g ∈M has order p, then Γg ⊂ Γ+0 (p) has
genus 0.
By Ogg, if p 6∈ Oggss , then X+0 (p) has positive genus.
![Page 38: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/38.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question B
If p 6∈ Oggss , then why do we expect p - #M?
Answer
By Moonshine, if g ∈M has order p, then Γg ⊂ Γ+0 (p) has
genus 0.
By Ogg, if p 6∈ Oggss , then X+0 (p) has positive genus.
![Page 39: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/39.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question B
If p 6∈ Oggss , then why do we expect p - #M?
Answer
By Moonshine, if g ∈M has order p, then Γg ⊂ Γ+0 (p) has
genus 0.
By Ogg, if p 6∈ Oggss , then X+0 (p) has positive genus.
![Page 40: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/40.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question C
If p ∈ Oggss , then why do we expect (a priori) that p | #M?
Heuristic Argument?
Let hp(τ) be the hauptmodul for Γ+0 (p).
Hecke implies that hp | U(p) ≡ (j − 744) | U(p) (mod p).
Deligne for Ep−1 gives hp | U(p) ∈ Sp−1(1) (mod p).
Implies j ′(hp | U(p)) ∈ Sp+1(1) (mod p).
Moonshine implies j ′(hp | U(p)) comes from Θ’s.
But Serre implies j ′(hp | U(p)) ∈ S2(p) (mod p).
We expect S2(p) (mod p) to be spanned by Θ′s.
Pizer proved Θ′s from quaternion alg’s suffice iff p ∈ Oggss .
![Page 41: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/41.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question C
If p ∈ Oggss , then why do we expect (a priori) that p | #M?
Heuristic Argument?
Let hp(τ) be the hauptmodul for Γ+0 (p).
Hecke implies that hp | U(p) ≡ (j − 744) | U(p) (mod p).
Deligne for Ep−1 gives hp | U(p) ∈ Sp−1(1) (mod p).
Implies j ′(hp | U(p)) ∈ Sp+1(1) (mod p).
Moonshine implies j ′(hp | U(p)) comes from Θ’s.
But Serre implies j ′(hp | U(p)) ∈ S2(p) (mod p).
We expect S2(p) (mod p) to be spanned by Θ′s.
Pizer proved Θ′s from quaternion alg’s suffice iff p ∈ Oggss .
![Page 42: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/42.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question C
If p ∈ Oggss , then why do we expect (a priori) that p | #M?
Heuristic Argument?
Let hp(τ) be the hauptmodul for Γ+0 (p).
Hecke implies that hp | U(p) ≡ (j − 744) | U(p) (mod p).
Deligne for Ep−1 gives hp | U(p) ∈ Sp−1(1) (mod p).
Implies j ′(hp | U(p)) ∈ Sp+1(1) (mod p).
Moonshine implies j ′(hp | U(p)) comes from Θ’s.
But Serre implies j ′(hp | U(p)) ∈ S2(p) (mod p).
We expect S2(p) (mod p) to be spanned by Θ′s.
Pizer proved Θ′s from quaternion alg’s suffice iff p ∈ Oggss .
![Page 43: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/43.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question C
If p ∈ Oggss , then why do we expect (a priori) that p | #M?
Heuristic Argument?
Let hp(τ) be the hauptmodul for Γ+0 (p).
Hecke implies that hp | U(p) ≡ (j − 744) | U(p) (mod p).
Deligne for Ep−1 gives hp | U(p) ∈ Sp−1(1) (mod p).
Implies j ′(hp | U(p)) ∈ Sp+1(1) (mod p).
Moonshine implies j ′(hp | U(p)) comes from Θ’s.
But Serre implies j ′(hp | U(p)) ∈ S2(p) (mod p).
We expect S2(p) (mod p) to be spanned by Θ′s.
Pizer proved Θ′s from quaternion lag’s suffice iff p ∈ Oggss .
![Page 44: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/44.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question C
If p ∈ Oggss , then why do we expect (a priori) that p | #M?
Heuristic Argument?
Let hp(τ) be the hauptmodul for Γ+0 (p).
Hecke implies that hp | U(p) ≡ (j − 744) | U(p) (mod p).
Deligne for Ep−1 gives hp | U(p) ∈ Sp−1(1) (mod p).
Implies j ′(hp | U(p)) ∈ Sp+1(1) (mod p).
Moonshine implies j ′(hp | U(p)) comes from Θ’s.
But Serre implies j ′(hp | U(p)) ∈ S2(p) (mod p).
We expect S2(p) (mod p) to be spanned by Θ′s.
Pizer proved Θ′s from quaternion alg’s suffice iff p ∈ Oggss .
![Page 45: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/45.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question C
If p ∈ Oggss , then why do we expect (a priori) that p | #M?
Heuristic Argument?
Let hp(τ) be the hauptmodul for Γ+0 (p).
Hecke implies that hp | U(p) ≡ (j − 744) | U(p) (mod p).
Deligne for Ep−1 gives hp | U(p) ∈ Sp−1(1) (mod p).
Implies j ′(hp | U(p)) ∈ Sp+1(1) (mod p).
Moonshine “expects” j ′(hp | U(p)) to come from Θ’s.
But Serre implies j ′(hp | U(p)) ∈ S2(p) (mod p).
We expect S2(p) (mod p) to be spanned by Θ′s.
Pizer proved Θ′s from quaternion alg’s suffice iff p ∈ Oggss .
![Page 46: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/46.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question C
If p ∈ Oggss , then why do we expect (a priori) that p | #M?
Heuristic Argument?
Let hp(τ) be the hauptmodul for Γ+0 (p).
Hecke implies that hp | U(p) ≡ (j − 744) | U(p) (mod p).
Deligne for Ep−1 gives hp | U(p) ∈ Sp−1(1) (mod p).
Implies j ′(hp | U(p)) ∈ Sp+1(1) (mod p).
Moonshine “expects” j ′(hp | U(p)) to come from Θ’s.
But Serre implies j ′(hp | U(p)) ∈ S2(p) (mod p).
We expect S2(p) (mod p) to be spanned by Θ′s.
Pizer proved Θ′s from quaternion alg’s suffice iff p ∈ Oggss .
![Page 47: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/47.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question C
If p ∈ Oggss , then why do we expect (a priori) that p | #M?
Heuristic Argument?
Let hp(τ) be the hauptmodul for Γ+0 (p).
Hecke implies that hp | U(p) ≡ (j − 744) | U(p) (mod p).
Deligne for Ep−1 gives hp | U(p) ∈ Sp−1(1) (mod p).
Implies j ′(hp | U(p)) ∈ Sp+1(1) (mod p).
Moonshine “expects” j ′(hp | U(p)) to come from Θ’s.
But Serre implies j ′(hp | U(p)) ∈ S2(p) (mod p).
We expect S2(p) (mod p) to be spanned by Θ′s.
Pizer proved Θ′s from quaternion alg’s suffice iff p ∈ Oggss .
![Page 48: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/48.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Ogg’s Problem
Question C
If p ∈ Oggss , then why do we expect (a priori) that p | #M?
Heuristic Argument?
Let hp(τ) be the hauptmodul for Γ+0 (p).
Hecke implies that hp | U(p) ≡ (j − 744) | U(p) (mod p).
Deligne for Ep−1 gives hp | U(p) ∈ Sp−1(1) (mod p).
Implies j ′(hp | U(p)) ∈ Sp+1(1) (mod p).
Moonshine “expects” j ′(hp | U(p)) to come from Θ’s.
But Serre implies j ′(hp | U(p)) ∈ S2(p) (mod p).
We expect S2(p) (mod p) to be spanned by Θ′s.
Pizer proved Θ′s from quaternion alg’s suffice iff p ∈ Oggss .
![Page 49: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/49.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Witten’s Conjecture (2007)
Conjecture (Witten, Li-Song-Strominger)
The vertex operator algebra V \ is dual to a 3d quantum gravitytheory. Thus, there are 194 “black hole states”.
Question (Witten)
How are these different kinds of black hole states distributed?
![Page 50: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/50.jpg)
Moonshine: Lecture 3
The Jack Daniels Problem
Witten’s Conjecture (2007)
Conjecture (Witten, Li-Song-Strominger)
The vertex operator algebra V \ is dual to a 3d quantum gravitytheory. Thus, there are 194 “black hole states”.
Question (Witten)
How are these different kinds of black hole states distributed?
![Page 51: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/51.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Distribution of Monstrous Moonshine
![Page 52: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/52.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Open Problem
Question
Consider the moonshine expressions
196884 = 1 + 196883
21493760 = 1 + 196883 + 21296876
864299970 = 1 + 1 + 196883 + 196883 + 21296876 + 842609326...
c(n) =194∑i=1
mi (n)χi (e)
How many ‘1’s, ‘196883’s, etc. show up in these equations?
![Page 53: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/53.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Some Proportions
n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))
-1 1 0 · · · 01 1/2 1/2 · · · 0...
......
......
40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .
60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .
100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .
![Page 54: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/54.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Some Proportions
n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))
-1 1 0 · · · 01 1/2 1/2 · · · 0...
......
......
40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .
80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .
![Page 55: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/55.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Some Proportions
n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))
-1 1 0 · · · 01 1/2 1/2 · · · 0...
......
......
40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .
100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .
160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .
![Page 56: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/56.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Some Proportions
n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))
-1 1 0 · · · 01 1/2 1/2 · · · 0...
......
......
40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .
100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .
220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .
![Page 57: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/57.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Some Proportions
n δ (m1(n)) δ (m2(n)) · · · δ (m194(n))
-1 1 0 · · · 01 1/2 1/2 · · · 0...
......
......
40 4.011 . . .× 10−4 2.514 . . .× 10−3 · · · 0.00891. . .60 2.699 . . .× 10−9 2.732 . . .× 10−8 · · · 0.04419. . .80 4.809 . . .× 10−14 7.537 . . .× 10−13 · · · 0.04428. . .
100 4.427 . . .× 10−18 1.077 . . .× 10−16 · · · 0.04428. . .120 1.377 . . .× 10−21 5.501 . . .× 10−20 · · · 0.04428. . .140 1.156 . . .× 10−24 1.260 . . .× 10−22 · · · 0.04428. . .160 2.621 . . .× 10−27 3.443 . . .× 10−23 · · · 0.04428. . .180 1.877 . . .× 10−28 3.371 . . .× 10−23 · · · 0.04428. . .200 1.715 . . .× 10−28 3.369 . . .× 10−23 · · · 0.04428. . .220 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .240 1.711 . . .× 10−28 3.368 . . .× 10−23 · · · 0.04428. . .
![Page 58: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/58.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Distribution of Moonshine
Theorem 1 (Duncan, Griffin, O)
We have Rademacher style exact formulas for mi (n) of the form
mi (n) =∑χi
∑c
Kloosterman sums× I -Bessel fcns
Remark
The dominant term gives
mi (n) ∼ dim(χi )√2|n|3/4|M|
· e4π√|n|
![Page 59: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/59.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Distribution of Moonshine
Theorem 1 (Duncan, Griffin, O)
We have Rademacher style exact formulas for mi (n) of the form
mi (n) =∑χi
∑c
Kloosterman sums× I -Bessel fcns
Remark
The dominant term gives
mi (n) ∼ dim(χi )√2|n|3/4|M|
· e4π√|n|
![Page 60: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/60.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Distribution
Remark
We have that
δ (mi ) := limn→+∞
mi (n)∑194i=1 mi (n)
is well defined
, and
δ (mi ) =dim(χi )∑194j=1 dim(χj)
=dim(χi )
5844076785304502808013602136.
![Page 61: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/61.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Distribution
Remark
We have that
δ (mi ) := limn→+∞
mi (n)∑194i=1 mi (n)
is well defined, and
δ (mi ) =dim(χi )∑194j=1 dim(χj)
=dim(χi )
5844076785304502808013602136.
![Page 62: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/62.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Proof
Orthogonality
Fact
If G is a group and g , h ∈ G , then
∑χi
χi (g)χi (h) =
{|CG (g)| If g and h are conjugate
0 otherwise,
where CG (g) is the centralizer of g in G .
![Page 63: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/63.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Proof
Tχ(τ)
Define
Tχi (τ) =1
|M|∑g∈M
χi (g)Tg (τ).
The orthogonality of characters gives the inverse relation
Tg (τ) =194∑i=1
χi (g)Tχi (τ).
From this we can work out that
Tχi (τ) =∞∑
n=−1
mi (n)qn.
![Page 64: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/64.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Proof
Tχ(τ)
Define
Tχi (τ) =1
|M|∑g∈M
χi (g)Tg (τ).
The orthogonality of characters gives the inverse relation
Tg (τ) =194∑i=1
χi (g)Tχi (τ).
From this we can work out that
Tχi (τ) =∞∑
n=−1
mi (n)qn.
![Page 65: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/65.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Proof
Tχ(τ)
Define
Tχi (τ) =1
|M|∑g∈M
χi (g)Tg (τ).
The orthogonality of characters gives the inverse relation
Tg (τ) =194∑i=1
χi (g)Tχi (τ).
From this we can work out that
Tχi (τ) =∞∑
n=−1
mi (n)qn.
![Page 66: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/66.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Proof
Outline of the proof
Theorem 2 (Duncan, Griffin,O)
We have exact formulas for the coeffs of the Tg (τ) and Tχi (τ).
Sketch of Proof
1 Each Γg contains some congruence subgroup Γ0(Ng ).
2 We first find the poles of Tg (τ).
3 We can then build a form with matching poles via Poincareseries.
4 The Poincare series give exact formulas for coefficients.
![Page 67: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/67.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Proof
Outline of the proof
Theorem 2 (Duncan, Griffin,O)
We have exact formulas for the coeffs of the Tg (τ) and Tχi (τ).
Sketch of Proof
1 Each Γg contains some congruence subgroup Γ0(Ng ).
2 We first find the poles of Tg (τ).
3 We can then build a form with matching poles via Poincareseries.
4 The Poincare series give exact formulas for coefficients.
![Page 68: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/68.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Proof
Outline of the proof
Theorem 2 (Duncan, Griffin,O)
We have exact formulas for the coeffs of the Tg (τ) and Tχi (τ).
Sketch of Proof
1 Each Γg contains some congruence subgroup Γ0(Ng ).
2 We first find the poles of Tg (τ).
3 We can then build a form with matching poles via Poincareseries.
4 The Poincare series give exact formulas for coefficients.
![Page 69: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/69.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Proof
Outline of the proof
Theorem 2 (Duncan, Griffin,O)
We have exact formulas for the coeffs of the Tg (τ) and Tχi (τ).
Sketch of Proof
1 Each Γg contains some congruence subgroup Γ0(Ng ).
2 We first find the poles of Tg (τ).
3 We can then build a form with matching poles via Poincareseries.
4 The Poincare series give exact formulas for coefficients.
![Page 70: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/70.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Proof
Outline of the proof
Theorem 2 (Duncan, Griffin,O)
We have exact formulas for the coeffs of the Tg (τ) and Tχi (τ).
Sketch of Proof
1 Each Γg contains some congruence subgroup Γ0(Ng ).
2 We first find the poles of Tg (τ).
3 We can then build a form with matching poles via Poincareseries.
4 The Poincare series give exact formulas for coefficients.
![Page 71: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/71.jpg)
Moonshine: Lecture 3
II. Distribution of Monstrous Moonshine
Proof
Outline of the proof
Theorem 2 (Duncan, Griffin,O)
We have exact formulas for the coeffs of the Tg (τ) and Tχi (τ).
Sketch of Proof
1 Each Γg contains some congruence subgroup Γ0(Ng ).
2 We first find the poles of Tg (τ).
3 We can then build a form with matching poles via Poincareseries.
4 The Poincare series give exact formulas for coefficients.
![Page 72: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/72.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Umbral (shadow) Moonshine
![Page 73: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/73.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Present day moonshine
Observation (Eguchi, Ooguri, Tachikawa (2010))
There is a mock modular form
H(τ) = q−18(−2 + 45q + 231q2 + 770q3 + 2277q4 + 5796q5 + ...
).
The degrees of the irreducible repn’s of the Mathieu group M24 are:
1, 23,45, 231, 252, 253, 483, 770, 990, 1035,
1265, 1771, 2024, 2277, 3312, 3520, 5313, 5544, 5796, 10395.
![Page 74: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/74.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Present day moonshine
Observation (Eguchi, Ooguri, Tachikawa (2010))
There is a mock modular form
H(τ) = q−18(−2 + 45q + 231q2 + 770q3 + 2277q4 + 5796q5 + ...
).
The degrees of the irreducible repn’s of the Mathieu group M24 are:
1, 23,45, 231, 252, 253, 483, 770, 990, 1035,
1265, 1771, 2024, 2277, 3312, 3520, 5313, 5544, 5796, 10395.
![Page 75: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/75.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mathieu Moonshine
Theorem (Gannon (2013))
There is an infinite dimensional graded M24-module whoseMcKay-Thompson series are specific mock modular forms.
Remark
Alleged multiplicities must be integral and non-negative.
Computed using wgt 1/2 weakly holomorphic modular forms.
Integrality follows from “theory of modular forms mod p”.
Non-negativity follows from “effectivizing” argument ofBringmann-O on Ramanujan’s f (q) mock theta function.
![Page 76: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/76.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mathieu Moonshine
Theorem (Gannon (2013))
There is an infinite dimensional graded M24-module whoseMcKay-Thompson series are specific mock modular forms.
Remark
Alleged multiplicities must be integral and non-negative.
Computed using wgt 1/2 weakly holomorphic modular forms.
Integrality follows from “theory of modular forms mod p”.
Non-negativity follows from “effectivizing” argument ofBringmann-O on Ramanujan’s f (q) mock theta function.
![Page 77: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/77.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mathieu Moonshine
Theorem (Gannon (2013))
There is an infinite dimensional graded M24-module whoseMcKay-Thompson series are specific mock modular forms.
Remark
Alleged multiplicities must be integral and non-negative.
Computed using wgt 1/2 weakly holomorphic modular forms.
Integrality follows from “theory of modular forms mod p”.
Non-negativity follows from “effectivizing” argument ofBringmann-O on Ramanujan’s f (q) mock theta function.
![Page 78: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/78.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mathieu Moonshine
Theorem (Gannon (2013))
There is an infinite dimensional graded M24-module whoseMcKay-Thompson series are specific mock modular forms.
Remark
Alleged multiplicities must be integral and non-negative.
Computed using wgt 1/2 weakly holomorphic modular forms.
Integrality follows from “theory of modular forms mod p”.
Non-negativity follows from “effectivizing” argument ofBringmann-O on Ramanujan’s f (q) mock theta function.
![Page 79: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/79.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mathieu Moonshine
Theorem (Gannon (2013))
There is an infinite dimensional graded M24-module whoseMcKay-Thompson series are specific mock modular forms.
Remark
Alleged multiplicities must be integral and non-negative.
Computed using wgt 1/2 weakly holomorphic modular forms.
Integrality follows from “theory of modular forms mod p”.
Non-negativity follows from “effectivizing” argument ofBringmann-O on Ramanujan’s f (q) mock theta function.
![Page 80: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/80.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
What are mock modular forms?
Notation. Throughout, let
τ = x + iy ∈ H with x , y ∈ R.
Hyperbolic Laplacian.
∆k := −y2
(∂2
∂x2+
∂2
∂y2
)+ iky
(∂
∂x+ i
∂
∂y
).
![Page 81: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/81.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
What are mock modular forms?
Notation. Throughout, let
τ = x + iy ∈ H with x , y ∈ R.
Hyperbolic Laplacian.
∆k := −y2
(∂2
∂x2+
∂2
∂y2
)+ iky
(∂
∂x+ i
∂
∂y
).
![Page 82: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/82.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Harmonic Maass forms
Definition
A harmonic Maass form of weight k on a subgroup Γ ⊂ SL2(Z) isany smooth function M : H→ C satisfying:
1 For all A =(a bc d
)∈ Γ and z ∈ H, we have
M
(aτ + b
cτ + d
)= (cz + d)k M(τ).
2 We have that ∆kM = 0.
![Page 83: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/83.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Harmonic Maass forms
Definition
A harmonic Maass form of weight k on a subgroup Γ ⊂ SL2(Z) isany smooth function M : H→ C satisfying:
1 For all A =(a bc d
)∈ Γ and z ∈ H, we have
M
(aτ + b
cτ + d
)= (cz + d)k M(τ).
2 We have that ∆kM = 0.
![Page 84: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/84.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Harmonic Maass forms
Definition
A harmonic Maass form of weight k on a subgroup Γ ⊂ SL2(Z) isany smooth function M : H→ C satisfying:
1 For all A =(a bc d
)∈ Γ and z ∈ H, we have
M
(aτ + b
cτ + d
)= (cz + d)k M(τ).
2 We have that ∆kM = 0.
![Page 85: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/85.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Fourier expansions
Fundamental Lemma
If M ∈ H2−k and Γ(a, x) is the incomplete Γ-function, then
M(τ) =∑
n�−∞c+(n)qn +
∑n<0
c−(n)Γ(k − 1, 4π|n|y)qn.
l lHolomorphic part M+ Nonholomorphic part M−
Remark
We call M+ a mock modular form.
If ξ2−k := 2iy2−k ∂∂τ , then the shadow of M is ξ2−k(M−).
![Page 86: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/86.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Fourier expansions
Fundamental Lemma
If M ∈ H2−k and Γ(a, x) is the incomplete Γ-function, then
M(τ) =∑
n�−∞c+(n)qn +
∑n<0
c−(n)Γ(k − 1, 4π|n|y)qn.
l lHolomorphic part M+ Nonholomorphic part M−
Remark
We call M+ a mock modular form.
If ξ2−k := 2iy2−k ∂∂τ , then the shadow of M is ξ2−k(M−).
![Page 87: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/87.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Fourier expansions
Fundamental Lemma
If M ∈ H2−k and Γ(a, x) is the incomplete Γ-function, then
M(τ) =∑
n�−∞c+(n)qn +
∑n<0
c−(n)Γ(k − 1, 4π|n|y)qn.
l lHolomorphic part M+ Nonholomorphic part M−
Remark
We call M+ a mock modular form.
If ξ2−k := 2iy2−k ∂∂τ , then the shadow of M is ξ2−k(M−).
![Page 88: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/88.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Fourier expansions
Fundamental Lemma
If M ∈ H2−k and Γ(a, x) is the incomplete Γ-function, then
M(τ) =∑
n�−∞c+(n)qn +
∑n<0
c−(n)Γ(k − 1, 4π|n|y)qn.
l lHolomorphic part M+ Nonholomorphic part M−
Remark
We call M+ a mock modular form.
If ξ2−k := 2iy2−k ∂∂τ , then the shadow of M is ξ2−k(M−).
![Page 89: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/89.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Shadows are modular forms
Fundamental Lemma
The operator ξ2−k := 2iy2−k ∂∂τ defines a surjective map
ξ2−k : H2−k −→ Sk .
Remark
In M24 Moonshine, the McKay-Thompson series are mock modularforms with classical Jacobi theta series shadows!
![Page 90: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/90.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Shadows are modular forms
Fundamental Lemma
The operator ξ2−k := 2iy2−k ∂∂τ defines a surjective map
ξ2−k : H2−k −→ Sk .
Remark
In M24 Moonshine, the McKay-Thompson series are mock modularforms with classical Jacobi theta series shadows!
![Page 91: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/91.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Larger Framework of Moonshine?
Remark
There are well known connections with even unimodular positivedefinite rank 24 lattices:
M ←→ Leech lattice
M24 ←→ A241 lattice.
![Page 92: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/92.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Umbral Moonshine Conjecture
Conjecture (Cheng, Duncan, Harvey (2013))
Let LX (up to isomorphism) be an even unimodularpositive-definite rank 24 lattice, and let :
X be the corresponding ADE-type root system.
W X the Weyl group of X .
The umbral group GX := Aut(LX )/W X .
For each g ∈ GX let HXg (τ) be a specific automorphic form
with minimal principal parts.
Then there is an infinite dimensional graded GX module KX forwhich HX
g (τ) is the McKay-Thompson series for g .
![Page 93: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/93.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Umbral Moonshine Conjecture
Conjecture (Cheng, Duncan, Harvey (2013))
Let LX (up to isomorphism) be an even unimodularpositive-definite rank 24 lattice, and let :
X be the corresponding ADE-type root system.
W X the Weyl group of X .
The umbral group GX := Aut(LX )/W X .
For each g ∈ GX let HXg (τ) be a specific automorphic form
with minimal principal parts.
Then there is an infinite dimensional graded GX module KX forwhich HX
g (τ) is the McKay-Thompson series for g .
![Page 94: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/94.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Umbral Moonshine Conjecture
Conjecture (Cheng, Duncan, Harvey (2013))
Let LX (up to isomorphism) be an even unimodularpositive-definite rank 24 lattice, and let :
X be the corresponding ADE-type root system.
W X the Weyl group of X .
The umbral group GX := Aut(LX )/W X .
For each g ∈ GX let HXg (τ) be a specific automorphic form
with minimal principal parts.
Then there is an infinite dimensional graded GX module KX forwhich HX
g (τ) is the McKay-Thompson series for g .
![Page 95: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/95.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Umbral Moonshine Conjecture
Conjecture (Cheng, Duncan, Harvey (2013))
Let LX (up to isomorphism) be an even unimodularpositive-definite rank 24 lattice, and let :
X be the corresponding ADE-type root system.
W X the Weyl group of X .
The umbral group GX := Aut(LX )/W X .
For each g ∈ GX let HXg (τ) be a specific automorphic form
with minimal principal parts.
Then there is an infinite dimensional graded GX module KX forwhich HX
g (τ) is the McKay-Thompson series for g .
![Page 96: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/96.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Umbral Moonshine Conjecture
Conjecture (Cheng, Duncan, Harvey (2013))
Let LX (up to isomorphism) be an even unimodularpositive-definite rank 24 lattice, and let :
X be the corresponding ADE-type root system.
W X the Weyl group of X .
The umbral group GX := Aut(LX )/W X .
For each g ∈ GX let HXg (τ) be a specific automorphic form
with minimal principal parts.
Then there is an infinite dimensional graded GX module KX forwhich HX
g (τ) is the McKay-Thompson series for g .
![Page 97: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/97.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Remarks
Cheng, Duncan and Harvey constructed their mock modularforms using “Rademacher sums”.
For X = A122 we have GX = M24 and Gannon’s Theorem.
There are 22 other isomorphism classes of X , the HXg (τ)
constructed from X and its Coxeter number m(X ).
Remark
Apart from the Leech case, the HXg (τ) are always weight 1/2 mock
modular forms whose shadows are weight 3/2 cuspidal theta serieswith level m(X ).
![Page 98: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/98.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Remarks
Cheng, Duncan and Harvey constructed their mock modularforms using “Rademacher sums”.
For X = A122 we have GX = M24 and Gannon’s Theorem.
There are 22 other isomorphism classes of X , the HXg (τ)
constructed from X and its Coxeter number m(X ).
Remark
Apart from the Leech case, the HXg (τ) are always weight 1/2 mock
modular forms whose shadows are weight 3/2 cuspidal theta serieswith level m(X ).
![Page 99: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/99.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Remarks
Cheng, Duncan and Harvey constructed their mock modularforms using “Rademacher sums”.
For X = A122 we have GX = M24 and Gannon’s Theorem.
There are 22 other isomorphism classes of X , the HXg (τ)
constructed from X and its Coxeter number m(X ).
Remark
Apart from the Leech case, the HXg (τ) are always weight 1/2 mock
modular forms whose shadows are weight 3/2 cuspidal theta serieswith level m(X ).
![Page 100: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/100.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Remarks
Cheng, Duncan and Harvey constructed their mock modularforms using “Rademacher sums”.
For X = A122 we have GX = M24 and Gannon’s Theorem.
There are 22 other isomorphism classes of X , the HXg (τ)
constructed from X and its Coxeter number m(X ).
Remark
Apart from the Leech case, the HXg (τ) are always weight 1/2 mock
modular forms whose shadows are weight 3/2 cuspidal theta serieswith level m(X ).
![Page 101: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/101.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Remarks
Cheng, Duncan and Harvey constructed their mock modularforms using “Rademacher sums”.
For X = A122 we have GX = M24 and Gannon’s Theorem.
There are 22 other isomorphism classes of X , the HXg (τ)
constructed from X and its Coxeter number m(X ).
Remark
Apart from the Leech case, the HXg (τ) are always weight 1/2 mock
modular forms whose shadows are weight 3/2 cuspidal theta serieswith level m(X ).
![Page 102: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/102.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Our results....
Theorem 3 (Duncan, Griffin, Ono)
The Umbral Moonshine Conjecture is true.
Remark
This result is a “numerical proof” of Umbral moonshine. It isanalogous to the work of Atkin, Fong and Smith in the case ofmonstrous moonshine.
![Page 103: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/103.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Our results....
Theorem 3 (Duncan, Griffin, Ono)
The Umbral Moonshine Conjecture is true.
Remark
This result is a “numerical proof” of Umbral moonshine. It isanalogous to the work of Atkin, Fong and Smith in the case ofmonstrous moonshine.
![Page 104: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/104.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Our results....
Theorem 3 (Duncan, Griffin, Ono)
The Umbral Moonshine Conjecture is true.
Remark
This result is a “numerical proof” of Umbral moonshine. It isanalogous to the work of Atkin, Fong and Smith in the case ofmonstrous moonshine.
![Page 105: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/105.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Beautiful examples
Example
For M12 the MT series include Ramanujan’s mock thetas:
f (q) = 1 +∞∑n=1
qn2
(1 + q)2(1 + q2)2 · · · (1 + qn)2,
φ(q) = 1 +∞∑n=1
qn2
(1 + q2)(1 + q4) · · · (1 + q2n),
χ(q) = 1 +∞∑n=1
qn2
(1− q + q2)(1− q2 + q4) · · · (1− qn + q2n)
![Page 106: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/106.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Mock modular forms
Beautiful examples
Example
For M12 the MT series include Ramanujan’s mock thetas:
f (q) = 1 +∞∑n=1
qn2
(1 + q)2(1 + q2)2 · · · (1 + qn)2,
φ(q) = 1 +∞∑n=1
qn2
(1 + q2)(1 + q4) · · · (1 + q2n),
χ(q) = 1 +∞∑n=1
qn2
(1− q + q2)(1− q2 + q4) · · · (1− qn + q2n)
![Page 107: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/107.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Strategy of Proof
For each X we compute non-negative integers mXi (n) for which
KX =∞∑
n=−1
∑χi
mXi (n)Vχi .
![Page 108: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/108.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Strategy of Proof
For each X we compute non-negative integers mXi (n) for which
KX =∞∑
n=−1
∑χi
mXi (n)Vχi .
![Page 109: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/109.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Orthogonality
Fact
If G is a group and g , h ∈ G , then
∑χi
χi (g)χi (h) =
{|CG (g)| If g and h are conjugate
0 otherwise,
where CG (g) is the centralizer of g in G .
![Page 110: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/110.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
TXχ (τ)
Define the weight 1/2 harmonic Maass form
TXχi
(τ) :=1
|M|∑g∈GX
χi (g)HXg (τ).
The orthogonality of characters gives the inverse relation
HXg (τ) =
∑χi (g)TX
χi(τ).
It would be great if
TXχi
(τ) =∞∑
n=−1
mXi (n)qn.
We only need to establish integrality and non-negativity!
![Page 111: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/111.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
TXχ (τ)
Define the weight 1/2 harmonic Maass form
TXχi
(τ) :=1
|M|∑g∈GX
χi (g)HXg (τ).
The orthogonality of characters gives the inverse relation
HXg (τ) =
∑χi (g)TX
χi(τ).
It would be great if
TXχi
(τ) =∞∑
n=−1
mXi (n)qn.
We only need to establish integrality and non-negativity!
![Page 112: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/112.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
TXχ (τ)
Define the weight 1/2 harmonic Maass form
TXχi
(τ) :=1
|M|∑g∈GX
χi (g)HXg (τ).
The orthogonality of characters gives the inverse relation
HXg (τ) =
∑χi (g)TX
χi(τ).
It would be great if
TXχi
(τ) =∞∑
n=−1
mXi (n)qn.
We only need to establish integrality and non-negativity!
![Page 113: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/113.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
TXχ (τ)
Define the weight 1/2 harmonic Maass form
TXχi
(τ) :=1
|M|∑g∈GX
χi (g)HXg (τ).
The orthogonality of characters gives the inverse relation
HXg (τ) =
∑χi (g)TX
χi(τ).
It would be great if
TXχi
(τ) =∞∑
n=−1
mXi (n)qn.
We only need to establish integrality and non-negativity!
![Page 114: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/114.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Difficulties
We have that
TXχi
(τ) = “period integral of a Θ-function” +∞∑
n=−1
mXi (n)qn.
Method of holomorphic projection gives:
πhol : H 12−→ M2 = {wgt 2 quasimodular forms}.
![Page 115: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/115.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Difficulties
We have that
TXχi
(τ) = “period integral of a Θ-function” +∞∑
n=−1
mXi (n)qn.
Method of holomorphic projection gives:
πhol : H 12−→ M2 = {wgt 2 quasimodular forms}.
![Page 116: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/116.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Difficulties
We have that
TXχi
(τ) = “period integral of a Θ-function” +∞∑
n=−1
mXi (n)qn.
Method of holomorphic projection gives:
πhol : H 12−→ M2 = {wgt 2 quasimodular forms}.
![Page 117: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/117.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection
Definition
Let f be a wgt k ≥ 2 (not necessarily holomorphic) modular form
f (τ) =∑n∈Z
af (n, y)qn.
We say that f is admissible if the following hold:
1 At cusps we have f (γ−1w)(
ddw τ
) k2 = c0 + O(Im(w)−ε),
where w = γτ ,
2 af (n, y) = O(y2−k) as y → 0 for all n > 0.
![Page 118: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/118.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection
Definition
Let f be a wgt k ≥ 2 (not necessarily holomorphic) modular form
f (τ) =∑n∈Z
af (n, y)qn.
We say that f is admissible if the following hold:
1 At cusps we have f (γ−1w)(
ddw τ
) k2 = c0 + O(Im(w)−ε),
where w = γτ ,
2 af (n, y) = O(y2−k) as y → 0 for all n > 0.
![Page 119: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/119.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection
Definition
Let f be a wgt k ≥ 2 (not necessarily holomorphic) modular form
f (τ) =∑n∈Z
af (n, y)qn.
We say that f is admissible if the following hold:
1 At cusps we have f (γ−1w)(
ddw τ
) k2 = c0 + O(Im(w)−ε),
where w = γτ ,
2 af (n, y) = O(y2−k) as y → 0 for all n > 0.
![Page 120: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/120.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection
Definition
Let f be a wgt k ≥ 2 (not necessarily holomorphic) modular form
f (τ) =∑n∈Z
af (n, y)qn.
We say that f is admissible if the following hold:
1 At cusps we have f (γ−1w)(
ddw τ
) k2 = c0 + O(Im(w)−ε),
where w = γτ ,
2 af (n, y) = O(y2−k) as y → 0 for all n > 0.
![Page 121: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/121.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection
Definition
Let f be a wgt k ≥ 2 (not necessarily holomorphic) modular form
f (τ) =∑n∈Z
af (n, y)qn.
We say that f is admissible if the following hold:
1 At cusps we have f (γ−1w)(
ddw τ
) k2 = c0 + O(Im(w)−ε),
where w = γτ ,
2 af (n, y) = O(y2−k) as y → 0 for all n > 0.
![Page 122: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/122.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection continued
Definition
If f is an admissible wgt k ≥ 2 nonholomorphic modular form,then its holomorphic projection is
(πhol f )(τ) := (πkhol f )(τ) := c0 +∞∑n=1
c(n)qn,
where for n > 0 we have
c(n) =(4πn)k−1
(k − 2)!
∫ ∞0
af (n, y)e−4πnyyk−2dy .
![Page 123: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/123.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection continued
Definition
If f is an admissible wgt k ≥ 2 nonholomorphic modular form,then its holomorphic projection is
(πhol f )(τ) := (πkhol f )(τ) := c0 +∞∑n=1
c(n)qn,
where for n > 0 we have
c(n) =(4πn)k−1
(k − 2)!
∫ ∞0
af (n, y)e−4πnyyk−2dy .
![Page 124: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/124.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection continued
Definition
If f is an admissible wgt k ≥ 2 nonholomorphic modular form,then its holomorphic projection is
(πhol f )(τ) := (πkhol f )(τ) := c0 +∞∑n=1
c(n)qn,
where for n > 0 we have
c(n) =(4πn)k−1
(k − 2)!
∫ ∞0
af (n, y)e−4πnyyk−2dy .
![Page 125: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/125.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection continued
Fundamental Lemma
If f is an admissible wgt k ≥ 2 nonholomorphic modular form onΓ0(N), then the following are true.
1 If f is holomorphic, then πhol(f ) = f (τ).
2 The function πhol(f ) lies in the space Mk(Γ0(N)).
Remark
Holomorphic projections appeared earlier in works of Sturm, andGross-Zagier, and work of Imamoglu, Raum, and Richter, Mertens,and Zwegers in connection with mock modular forms.
![Page 126: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/126.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection continued
Fundamental Lemma
If f is an admissible wgt k ≥ 2 nonholomorphic modular form onΓ0(N), then the following are true.
1 If f is holomorphic, then πhol(f ) = f (τ).
2 The function πhol(f ) lies in the space Mk(Γ0(N)).
Remark
Holomorphic projections appeared earlier in works of Sturm, andGross-Zagier, and work of Imamoglu, Raum, and Richter, Mertens,and Zwegers in connection with mock modular forms.
![Page 127: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/127.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection continued
Fundamental Lemma
If f is an admissible wgt k ≥ 2 nonholomorphic modular form onΓ0(N), then the following are true.
1 If f is holomorphic, then πhol(f ) = f (τ).
2 The function πhol(f ) lies in the space Mk(Γ0(N)).
Remark
Holomorphic projections appeared earlier in works of Sturm, andGross-Zagier, and work of Imamoglu, Raum, and Richter, Mertens,and Zwegers in connection with mock modular forms.
![Page 128: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/128.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection continued
Fundamental Lemma
If f is an admissible wgt k ≥ 2 nonholomorphic modular form onΓ0(N), then the following are true.
1 If f is holomorphic, then πhol(f ) = f (τ).
2 The function πhol(f ) lies in the space Mk(Γ0(N)).
Remark
Holomorphic projections appeared earlier in works of Sturm, andGross-Zagier, and work of Imamoglu, Raum, and Richter, Mertens,and Zwegers in connection with mock modular forms.
![Page 129: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/129.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Holomorphic projection continued
Fundamental Lemma
If f is an admissible wgt k ≥ 2 nonholomorphic modular form onΓ0(N), then the following are true.
1 If f is holomorphic, then πhol(f ) = f (τ).
2 The function πhol(f ) lies in the space Mk(Γ0(N)).
Remark
Holomorphic projections appeared earlier in works of Sturm, andGross-Zagier, and work of Imamoglu, Raum, and Richter, Mertens,and Zwegers in connection with mock modular forms.
![Page 130: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/130.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Sketch of the proof of umbral moonshine
Compute each wgt 1/2 harmonic Maass form TXχi
(τ).
Up to the nonholomorphic part, they are the multiplicity genfcns for V X
χiof the alleged moonshine modules KX .
Compute holomorphic projections of products with shadows.
The mXχi
(n) are integers iff these holomorphic projectionssatisfy certain congruences.
The mXχi
(n) can be estimated using “infinite sums” ofKloosterman sums weighted by I -Bessel functions. Forsufficiently large n this establishes non-negativity.
Check the finitely many (less than 400) cases directly.
![Page 131: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/131.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Sketch of the proof of umbral moonshine
Compute each wgt 1/2 harmonic Maass form TXχi
(τ).
Up to the nonholomorphic part, they are the multiplicity genfcns for V X
χiof the alleged moonshine modules KX .
Compute holomorphic projections of products with shadows.
The mXχi
(n) are integers iff these holomorphic projectionssatisfy certain congruences.
The mXχi
(n) can be estimated using “infinite sums” ofKloosterman sums weighted by I -Bessel functions. Forsufficiently large n this establishes non-negativity.
Check the finitely many (less than 400) cases directly.
![Page 132: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/132.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Sketch of the proof of umbral moonshine
Compute each wgt 1/2 harmonic Maass form TXχi
(τ).
Up to the nonholomorphic part, they are the multiplicity genfcns for V X
χiof the alleged moonshine modules KX .
Compute holomorphic projections of products with shadows.
The mXχi
(n) are integers iff these holomorphic projectionssatisfy certain congruences.
The mXχi
(n) can be estimated using “infinite sums” ofKloosterman sums weighted by I -Bessel functions. Forsufficiently large n this establishes non-negativity.
Check the finitely many (less than 400) cases directly.
![Page 133: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/133.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Sketch of the proof of umbral moonshine
Compute each wgt 1/2 harmonic Maass form TXχi
(τ).
Up to the nonholomorphic part, they are the multiplicity genfcns for V X
χiof the alleged moonshine modules KX .
Compute holomorphic projections of products with shadows.
The mXχi
(n) are integers iff these holomorphic projectionssatisfy certain congruences.
The mXχi
(n) can be estimated using “infinite sums” ofKloosterman sums weighted by I -Bessel functions. Forsufficiently large n this establishes non-negativity.
Check the finitely many (less than 400) cases directly.
![Page 134: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/134.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Sketch of the proof of umbral moonshine
Compute each wgt 1/2 harmonic Maass form TXχi
(τ).
Up to the nonholomorphic part, they are the multiplicity genfcns for V X
χiof the alleged moonshine modules KX .
Compute holomorphic projections of products with shadows.
The mXχi
(n) are integers iff these holomorphic projectionssatisfy certain congruences.
The mXχi
(n) can be estimated using “infinite sums” ofKloosterman sums weighted by I -Bessel functions.
Forsufficiently large n this establishes non-negativity.
Check the finitely many (less than 400) cases directly.
![Page 135: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/135.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Sketch of the proof of umbral moonshine
Compute each wgt 1/2 harmonic Maass form TXχi
(τ).
Up to the nonholomorphic part, they are the multiplicity genfcns for V X
χiof the alleged moonshine modules KX .
Compute holomorphic projections of products with shadows.
The mXχi
(n) are integers iff these holomorphic projectionssatisfy certain congruences.
The mXχi
(n) can be estimated using “infinite sums” ofKloosterman sums weighted by I -Bessel functions. Forsufficiently large n this establishes non-negativity.
Check the finitely many (less than 400) cases directly.
![Page 136: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/136.jpg)
Moonshine: Lecture 3
III. Umbral Moonshine
Sketch of the proof
Sketch of the proof of umbral moonshine
Compute each wgt 1/2 harmonic Maass form TXχi
(τ).
Up to the nonholomorphic part, they are the multiplicity genfcns for V X
χiof the alleged moonshine modules KX .
Compute holomorphic projections of products with shadows.
The mXχi
(n) are integers iff these holomorphic projectionssatisfy certain congruences.
The mXχi
(n) can be estimated using “infinite sums” ofKloosterman sums weighted by I -Bessel functions. Forsufficiently large n this establishes non-negativity.
Check the finitely many (less than 400) cases directly.
![Page 137: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/137.jpg)
Moonshine: Lecture 3
Executive Summary
Distribution of Monstrous Moonshine
Theorem 1 (Duncan, Griffin, O)
If 1 ≤ i ≤ 194, then we have Rademacher style exact formulas formi (n).
Remark
We have that
δ (mi ) =dim(χi )∑194j=1 dim(χj)
=dim(χi )
5844076785304502808013602136.
![Page 138: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/138.jpg)
Moonshine: Lecture 3
Executive Summary
Distribution of Monstrous Moonshine
Theorem 1 (Duncan, Griffin, O)
If 1 ≤ i ≤ 194, then we have Rademacher style exact formulas formi (n).
Remark
We have that
δ (mi ) =dim(χi )∑194j=1 dim(χj)
=dim(χi )
5844076785304502808013602136.
![Page 139: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/139.jpg)
Moonshine: Lecture 3
Executive Summary
Umbral Moonshine
Theorem 3 (Duncan, Griffin, Ono)
The Umbral Moonshine Conjecture is true.
Question
Do the infinite dimensional graded GX modules KX exhibit deepstructure?Probably....and some work of Duncan and Harvey makes use ofindefinite theta series to obtain a VOA structure in the M24 case.
![Page 140: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/140.jpg)
Moonshine: Lecture 3
Executive Summary
Umbral Moonshine
Theorem 3 (Duncan, Griffin, Ono)
The Umbral Moonshine Conjecture is true.
Question
Do the infinite dimensional graded GX modules KX exhibit deepstructure?
Probably....and some work of Duncan and Harvey makes use ofindefinite theta series to obtain a VOA structure in the M24 case.
![Page 141: Moonshine: Lecture 3spring15/MaxPlanck3.pdfMoonshine: Lecture 3 I. History of Moonshine The Monster Conjecture (Fischer and Griess (1973)) There is a huge simple group (containing](https://reader034.vdocuments.mx/reader034/viewer/2022050313/5f757dc6e41dba5fba7db8b2/html5/thumbnails/141.jpg)
Moonshine: Lecture 3
Executive Summary
Umbral Moonshine
Theorem 3 (Duncan, Griffin, Ono)
The Umbral Moonshine Conjecture is true.
Question
Do the infinite dimensional graded GX modules KX exhibit deepstructure?Probably....and some work of Duncan and Harvey makes use ofindefinite theta series to obtain a VOA structure in the M24 case.