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Prime number races for elliptic curves over function fields. (Bias de Chebyshev pour les courbes elliptiques sur les corps de fonctions.) (English. French summary) Zbl 1367.11085
The classical prime number races concern the number of primes in given congruence classes. As was shown by M. Rubinstein and P. Sarnak [Exp. Math. 3, No. 3, 173–197 (1994; Zbl 0823.11050)], one can understand the situation tolerably well, under GRH together with an assumption on the linear independence of the ordinates of zeros of the relevant \(L\)-functions. B. Mazur [Bull. Am. Math. Soc., New Ser. 45, No. 2, 185–228 (2008; Zbl 1229.11001)] introduced a race for a given elliptic curve \(E\), in which one compares the count of primes for which \(a_p(E)\) is positive, against the count for primes where it is negative. This was investigated by Sarnak, who also discussed the race in which one merely looks at the sign of the sum function for \(a_p(E)/\sqrt{p}\), and showed that the analytic rank of \(E\) affects any bias. Again a version of RH and of a linear independence hypothesis are required.
The present paper concerns the analogue of this last problem over a function field. Specifically, one uses the function field \(K\) of a proper smooth geometrically connected curve over a finite field \(\mathbb{F}_q\) of characteristic at least 5. Let \(E/K\) be an elliptic curve with non-constant \(j\)-invariant. The race concerns the distribution of \[ T_E(X):=-\frac{X}{q^{X/2}}\sum_{\deg(v)\leq X}\frac{a_v}{q^{\deg(v)/2}}, \] where the sum over places \(v\) is restricted to \(v\) which are “good”.
The main result of the paper is then that \(T(X_E)\) has a limiting distribution, with mean \[ \frac{\sqrt{q}}{\sqrt{q}-1}\left(\text{rank}(E/K)-\frac{1}{2}\right). \] The variance is also given explicitly, in terms of zeros of the \(L\)-function of \(E/K\). Moreover, for any family of elliptic curves \(E/K\) of unbounded conductor satisfying a suitable linear independence assumption, and such that the rank does not grow too rapidly, a suitably normalized version of \(T_X(E)\) converges to the Gaussian distribution.
Further results concern families of quadratic twists, and the family of large-rank curves \(y^2+xy=x^3-t^d\), given by D. Ulmer [Ann. Math. (2) 155, No. 1, 295–315 (2002; Zbl 1109.11314)].

MSC:
11T55 Arithmetic theory of polynomial rings over finite fields
11M41 Other Dirichlet series and zeta functions
11N13 Primes in congruence classes
11G05 Elliptic curves over global fields
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