C. R. Acad. Sci. Paris, t. 329, S&ie I, p. 459-463, 1999

Theorie des nombres/Number Theory (CPomCtrie algCbrique/Algebraic Geometry)

On the Coleman-Chabauty bound

Kirti JOSHI, Pavlos TZERMIAS

Department of Mathematics, P.O. Box 210089, 617 N. Santa Rita, The University of Arizona, Tucson, AZ 85721-00X9, USA E-mail: {kirti,tzermias}@math.arizona.edu

(Re~u le 25 mai 1999, accept& aprAs &vision le 10 aoiit 1999)

Abstract. The Coleman-Chabauty bound is an upper bound for the number of rational points on a curve of genus ,q > 2 whose Jacobian has Mordell-Wcil rank T less than 9. The bound is given in terms of the genus of the curve and the number of F,,-points on the reduced curve, for all primes p of good reduction such that p > 2.9. In this Note we show that the hypothesis on the Mordell-Weil rank is essential. We do so by exhibiting, fur each prime I, > 5, an explicit family of cut-ves of genus (r~ - t)/Z (and rank at least (;u - 1)/2) for which the bound in question does not hold. Our examples show that the difference between the number of rational points and the bound in question can in fact be linear in the genus. Under mild assumptions, our curves have rank at least twice their genus. 0 1999 Academic des Science&ditions scientifiques et medicales Elsevier SAS

SW la borne de Colemwn-Chabwuty

R&urn& La borne de Coleman-Chabanty est une borne supe’rieure sur le nombre de points rationnels d’une courbe de genre g 2 2 dont la Jacobienne a un rang de Mordell-Weil r plus petit que g. La borne est .spe’ci$ee par le genre de la courbe et le nombre de points F,,-rationnels sur Ia courbe reduite pour tout nombre premier p de bonne reduction tel que p > 29. Darts cette Note, nous montrons que la condition sur le rang de Mordell- Weil est essentielle en construisant explicitement, pour chaque nombre premier 11 > 5, une famille de courbes de genre .q = (p - 1)/2 (et de rang T > (p - 1)/Z) pour laquelle la borne en question n’est pas valable. Nos exemples montrent que la d@rence entre le nombre de points rationnefs et la borne est une function du genre qui croft au mnins lineairemertt. En outre, en imposant de faibles conditions suppltmentaires, nous tnontrons que ces courbes ont un rang superieur ou &al au double de leur genre. 0 1999 AcadCmie des Sciences/Editions scientifiques et medicales Elsevier SAS

Version francaise abrkghe

Soient X/Q une courbe projective et lisse de genre 9 2 2 et J la Jacobienne de X. Par la suite nous consid&ons des courbes X telles que X(Q) # 0 et plongeons X L) d. Coleman a donnC une

Note pr&+entCe par Jean-Pierre SERRE.

0764.4442/99/03290459 0 1999 Acadtmie des Sciences/l?ditions scicntifiques et mCdicales Elsevier SAS.

Tous droits reserves. 459

K. Joshi, P. Tzermias

version effective du rCsultat de Chabauty. 11 a en effet montr6 que si rang(J(Q)) 5 g - 1 alors, pour tout nombre premier p 2 2g + 1 tel que X a une bonne rkduction (que nous noterons X) pour p, on a

#X(Q) F #z(F,) + 29 - 2. (“1

Coleman a aussi soulevk le problkme du comportement du nombre de points rationnels sur des courbes dont la Jacobienne a un rang superieur ou Cgal au genre. Dans cette Note, nous montrons que la condition sur le rang de Mordell-Weil est essentielle en construisant explicitement une infinit6 de familles de courbes dont les points rationnels ne satisfont pas (*). En imposant de faibles conditions supplCmentaires, nous montrons d’autre part que le rang de Mordell-Weil des courbes que nous construisons est au moins deux fois plus grand que leur genre.

Notre rksultat principal est le thCor&me suivant :

TH~OR~ME 1. - Soit p > 5 un nombre premier. Conside’rons la famille de courbes :

x: y2 = (x - ag)(x - al). . . (ST - ap-1) + p2d2. (“*)

06 d est un entier non nul et (LO,. . . i aP-l sont des entiers distincts deux-&deux modulo p. Alors :

I. aucune courbe de la famille ne satisfait & la relation (*) (en particulier, le rang de la cow-be est supe’rieur ou &gal ~3 son genre 9 = (p - 1)/2) ;

2. si le membre de droite de (**> est un polyncime irrkductible sur Q, alors chaque membre de la famille a un rang r tel que 7‘ > 29.

De plus, une modification de nos exemples (voir remarque 1.3) montre que

lim sup #X(Q) - #%W - (2.9 - 2) > 2, Y-a 9

Nous discutons aussi des exemples du type Mestre-Shioda.

1. Main results

Let X/Q be a smooth projective curve of genus 9 2 2 and let J be the Jacobian of X. In what follows we will assume that X(Q) # 0 and embed X 4 .I. Coleman [3] gave an effective version of a result of Chabauty [2]. He used his p-adic integration methods to show that if ran&(<](Q)) 5 g - 1 then, for any prime p 2 2g + 1 for which X has good reduction (denoted by X in the sequel) at p, we have

#X(Q) I #y(F,) + 29 - 2. (1.1)

Grant [4] has given an example of a curve of genus 2 and rank 1 for which the above bound is sharp. In [3], Coleman raises the question of the behaviour of the number of rational points on curves

whose Jacobians have rank at least equal to the genus. The purpose of this note is to exhibit explicit infinite families of curves (necessarily of rank at least equal to their genus) whose set of rational points does not satisfy (1 .I). Moreover, under mild additional assumptions, the Mordell-Weil rank of the curves we construct is in fact at least twice their genus. Furthermore, a modification of our arguments, suggested by William McCallum and carried out by Dan Madden (see Remark 1.3),

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On the Coleman-Chabauty bound

shows that the difference between the left and right-hand side of (1.1) can be at least as large as twice the genus of the curve.

It should be noted that in [3] Coleman considers a class of hyperelliptic curves whose Jacobians have rank at least equal to the genus; however, the exhibited rational points on the latter curves are not in general enough to violate (1.1) (see Remark 1.4 below).

Our main result is the following theorem:

THEOREM 1.1. - Let p 2 5 be any prime. Consider the family of curves given by

x: y2 = (x -- ao)(x - al). . . (x - ap-l) + p2d2, (1.2)

where d is any non-zero integer and ao, . . . , ap-l are any integers which are pairwise distinct modulo p. Then:

1. every curve in the above family fails to satisfy (1.1) (in particular, the rank of the curve is at least equal to its genus g = (p - 1)/2);

2. if the right-hand side of (1.2) is an irreducible polynomial over Q, then every curve in the above family has rank at least equal to 29.

Proof - We first prove part 1. Note that, by our hypothesis on a;, the reduction X of X modulo p is given by

X: y2 = xp - 5, (1.3)

which is nonsingular. Also note that p =I 2g + 1 > 29. Now every affine F,-rational point on X is

of the form (a,O), where a E F,, so #X(Fp) = p + 1 = 2g + 2. Thus the right-hand side of (1.1) is 2g + 2 + 2g - 2 = 49. On the other hand, #X(Q) > 4g + 3, since the point at infinity and the points P; = (ai, pd) and P,’ = (ai, -pd) are all in X(Q). Therefore, (1.1) is not satisfied. By [3], the Mordell-Weil rank of J over Q is at least equal to g. Note also that all the above Q-rational points on X reduce to Weierstrass points on X.

To prove part 2, assume that the right-hand side of (1.2) is irreducible over Q. Let Qi be the affine Weierstrass points on X and Ri the affine Weierstrass points on r?. Consider the Albanese embedding of X in J sending cc to 0. The images of the points Qi generate all the a-torsion on J. We claim that J[2](Q) is trivial. Suppose not. Note that J[2] is a 2g-dimensional vector space over Fa generated by the images of the 2g + 1 points Qi in J, so there exists a unique relation among these points, namely 29+1 ,F1 Qi = 0 (in the Jacobian J). Now if J[2](Q) is not trivial then, rearranging the &is if necessary,

we may assume that there exists an integer d 5 g such that in J we have

&Q~=~Q:, (1.4) i=l i=l

for all g E Gal&/Q). S ince d 5 g, we conclude that each 0 permutes {Qi, . . . , Qd}, which contradicts the irreducibility of the right-hand side of (1.2). Now let I be the subgroup generated by the images of the points Pi under the fixed Albanese embedding. We claim that Itors is trivial. We argue using the formal group law on J. As p is a prime of good reduction we have an exact sequence

0 - G(pZ,,) - J(Q,) - @,) - 0, (1.5)

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where G is the formal group associated to J over Z, and by [5] we know that G(pZ,) is torsion-free (since-p is an odd prime). So all the torsion in IY is contained in the torsion of J(Q,), and injects

into J(F,). Further as noted in our proof of part 1, the points p; reduce to the points R, modulo p, therefore the image of r under the reduction map is 2-torsion. In particular, any element of Itcrrs is killed by 2. But Itors is Q-rational, so it has to be trivial, since .1[2](Q) is trivial. Also, by the exact sequence (1.5) and our remarks on the reduction of the points Pi, it is easy to see that I/2r surjects onto .I(F,)[2]. Therefore, since rtors is trivial, we conclude that the points P; generate a free subgroup of .r(Q) f o rank 29. This concludes our proof. q

Remark 1.2. - Part 2 of Theorem 1.1 remains valid for the family of curves

x: y2 = (2; - ufj)(:c - ul) . . (x - a2g) + pQ2. (1.6)

where g is a positive integer, p is a prime such that p > 29 + 1, $ is a non-zero integer and (~a,. . ., uzy are pairwise distinct integers modulo p.

Remark 1.3. - William McCallum has pointed out that Part 1 of Theorem 1.1 remains valid (perhaps at the expense of getting a less explicit equation) if one replaces n by a suitable polynomial d(.r) of degree 5 (p - 1)/2. The following elegant construction of such a polynomial was given to us by Dan Madden. Choose integers hr,. . ., t)(p--l),z such that p” divides b; - CL, and b, - cl,i # 0. Consider the polynomial g(z) = (X - (L~)(z - cur). . . (z - n,,-r). W e can write !/(b,) = p2sit;, where s,, t, are in Z. Now choose rationals ui and w, such that

ZL, + 711j = si. ‘U, - w; = ti.

Then g(b;) + p2$ = p2$. Note that the Vandermonde determinant of the b;‘s is not divisible by p. Therefore, by Lagrange interpolation, we can choose a polynomial d(z) of degree (p - 1)/2 such that the coefficients of ct(zr) are p-integral and d(b;) = TX;, for 1 < i < (p - 1)/2. Therefore, we get p - 1 extra Q-rational points (b,!pvi) and (bi; -pu;) (in addition to the points (aj.pci(ci,)) and (nj, -pd(nj))) on the curve

x: y2 = (cc - a(J)(:I: - al) ‘. . (x - f+l) + p2d(:r)*.

In particular,

lirrlsnp #X(Q) - #‘(F,) - (2.9 - 2) > 2. -

g-m 9

Remark 1.4. - In [3] Coleman produced a different family of curves whose Jacobians have rank at least g. The key point of the proof in [3] is that the explicit points do not reduce to Weierstrass points modulo p. On the other hand, in our examples all the rational points we construct reduce to Weierstrass points modulo p.

2. Mestre-Shioda type examples

J.-F. Mestre and T. Shioda (see [ 11, [6] and [7]) have constructed examples of hyperelliptic curves of genus g over Q which have fairly large number of rational points. It is natural to ask if these examples also fail to satisfy (1 .l).

PROPOSITION 2.1. - The Mestre-Shioda type examples give curves which fail to .satiTfj ( 1.1) provided that these curves have a prime p of good reduction in the range 29 < p < 3,q + 6.

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The proof is trivial using the easy fact that a hyperelliptic curve over a finite field Fq can have at most 2q + 2 points defined over Fq. We ran a computer search with Mestre-Shioda type examples for genus two and three curves and found that such a prime frequently exists.

3. Genus one case

In this section we show that the method of proof of Theorem 1.1 can also be applied to the case of genus one curves.

THEOREM 3.1. - Let E be the elliptic curve y2 = (x - u)(x - a + l)(z - a + 2) + 9d2, where a, d E Z and d # 0. Then E has positive Mordell--Weil rank and, if d is odd, then rk(E(Q)) > 2.

Prooj - First note that E has good reduction modulo 3. From the exact sequence (1.5) it follows that the point P = [(a, 3d) - (a, -3d)] is of infinite order, so the rank is always positive. The proof of part 2 of Theorem 1.1 applies to this case mutatis mutandis, provided we check irreducibility of

f(x) = (x - u)(z - a + 1)(x - a + 2) + Yd’, (3.1)

if d is odd. By Gauss’ theorem, it suffices to show that f(z) does not have a root in Z. Now observe that for an integer z, the product (z - U)(X - a + 1)(x - a + 2) is even, while 9d2 is odd. 0

Acknowledgments. This work was done in the stimulating atmosphere of the Arizona Winter School (March 1999) organized by the Southwestern Center for Arithmetical Algebraic Geometry. We are especially indebted to William McCallum for his inspiring lectures on Coleman’s effective Chabauty method and for his comments on this work. Special thanks are due to Dan Madden for Remark 1.3. We thank Minhyong Kim, Dinesh Thakur and Doug Ulmer for numerous conversations and comments. We are also indebted to Jean-Pierre Serre for his suggestions on the exposition and we also thank Pierre Colmez for his comments and suggestions. We thank Giampaolo D’ Alessandro, Alain Goriely and Joceline Lega for all their help with French translation.

References

[1] Caporaso L., Harris J., Mazur B., How many rational points can a curve have?, The moduli space of curves (Texel Island, 1994) Progr. Math. 129, Birkhauser Boston, 1995, pp. 13-31.

[2] Chabauty C., Sur les points rationels des courbes algebriques de genre superieur a l’unite, C. R. Acad. Sci. Paris 212 (1941) 882-885.

[3] Coleman R., Effective Chabauty, Duke Math. J. 52 (3) (1985) 765-770. [4] Grant D., A curve for which Coleman’s effective Chabauty bound is sharp, Proc. Amer. Math. Sot. 122 (1) (1994) 3 17-3 19. ]5] Serre J.-P., Lie algebras and Lie groups, 1964 lectures given at Harvard University, Lect. Notes in Math. 1500,

Springer-Verlag. 1992. [6] Shioda T., Constructing curves of high rank via symmetry, Amer. J. Math. 120 (3) (1998) 551-556.

[7] Shioda T., Genus two curves over Q(t) with high rank, Comment. Math. Univ. St. Paul. 46 (1) (1997) 15-21.

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