Prime number races for elliptic curves over function fields

Cha, Byungchul; Fiorilli, Daniel; Jouve, Florent

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1239-1277.

Nous étudions une version du biais de Chebyshev pour les courbes elliptiques sur le corps de fonctions d'une courbe lisse, propre, et géométriquement irréductible. Il s'agit de l'analogue, dans le cas des corps de fonctions, de travaux de Mazur, Sarnak, et Fiorilli. Le cadre géométrique dans lequel on se place permet d'établir inconditionnellement des résultats qui, sur les corps de nombres, nécessitent de supposer l'hypothèse de Riemann ou la conjecture de simplicité généralisée pour les zéros des fonctions $L$ intervenant. On démontre notamment que, dans certaines familles naturelles de courbes elliptiques, il y a dissipation générique du biais lorsque le conducteur tend vers l'infini. La preuve s'appuie sur des résultats d'indépendance linéaire de zéros de fonctions $L$ qui font l'objet d'un autre article des mêmes auteurs. Nous étudions par ailleurs la famille de courbes elliptiques d'Ulmer, et nous montrons qu'elle se comporte de manière pathologique, comparée au cas générique. Divers biais sont mis en évidence pour cette famille, dont certains sont conjecturalement impossibles à réaliser dans le cas des corps de nombres.

We study the prime number race for elliptic curves over the function field of a proper, smooth and geometrically connected curve over a finite field. This constitutes a function field analogue of prior work by Mazur, Sarnak and the second author. In this geometric setting, we can prove unconditional results whose counterparts in the number field case are conditional on a Riemann Hypothesis and a linear independence hypothesis on the zeros of the implied $L$-functions. Notably we show that in certain natural families of elliptic curves, the bias generically dissipates as the conductor grows. This is achieved by proving a central limit theorem and combining it with generic linear independence results that will appear in a separate paper. Also we study in detail a particular family of elliptic curves that have been considered by Ulmer. In contrast to the generic case we show that the race exhibits very diverse outcomes, some of which are believed to be impossible in the number field setting. Such behaviors are possible in the function field case because the zeros of Hasse-Weil $L$-functions for those elliptic curves can be proven to be highly dependent among themselves, which is a very non generic situation.

Classification : 11N45, 11G05, 11G40; 11N80.
Keywords: Chebyshev's bias, elliptic curves over function fields, summatory function of traces of Frobenius, linear independence of zeros of $L$-functions.
Mot clés : Biais de Chebyshev, courbes elliptiques sur les corps de fonctions, fonction sommatoire de la trace de Frobenius, indépendance linéaire des zéros de fonctions $L$.

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