[Paper Review] Prime number races for elliptic curves over function fields

Byungchul Cha, Daniel Fiorilli+1 more|arXiv (Cornell University)|Feb 18, 2015

Analytic Number Theory Research18 references3 citations

TL;DR

This paper studies prime number races in elliptic curves over function fields, proving unconditional results on bias in Frobenius trace distributions. It establishes that in generic families, bias diminishes as conductor grows due to a central limit theorem and generic linear independence of L-function zeros, while in special families (e.g., Ulmer’s), strong, non-generic biases persist due to dependent zeros.

ABSTRACT

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.

Motivation & Objective

• To investigate prime number races for elliptic curves over function fields, extending prior number field work by Mazur, Sarnak, and Fiorilli.
• To understand how the distribution of Frobenius traces (positive vs. negative) exhibits bias, and how this depends on the analytic properties of Hasse-Weil L-functions.
• To determine whether highly biased races—impossible in the number field setting under standard hypotheses—can occur in the function field setting.
• To establish that in generic families, bias dissipates as conductor grows, contrasting with conditional results in number fields.
• To analyze Ulmer’s family of elliptic curves, where non-generic zero dependencies lead to persistent, extreme biases

Proposed method

• Use the function field analogue of the Rubinstein-Sarnak framework to quantify bias via logarithmic densities of sets where π(x; q, a) > π(x; q, b).
• Apply a central limit theorem for Frobenius traces in families of elliptic curves over function fields to show convergence of bias to 1/2.
• Leverage generic linear independence of nontrivial zeros of L-functions (proven in a companion work) to rule out persistent bias in generic families.
• Analyze the structure of the L-function zeros via the functional equation, identifying trivial relations (e.g., γ_j and 2π - γ_j) that lead to non-generic dependencies.
• Use bounds on the number of curves with nontrivial relations among zeros (via exponential sums and conductor growth) to estimate the density of non-generic cases.
• Combine these bounds with Corollary 4.6 to show that curves violating linear independence or having rank ≥2 are rare, implying generic bias dissipation.

Experimental results

Research questions

• RQ1 Can prime number races in elliptic curves over function fields exhibit strong, persistent bias, and if so, under what conditions?
• RQ2 How does the bias in Frobenius trace sign distributions behave as the conductor grows in generic families of elliptic curves over function fields?
• RQ3 Why do certain families (e.g., Ulmer’s) exhibit extreme biases not seen in the number field case, and what structural properties of their L-functions enable this?
• RQ4 To what extent do dependencies among the zeros of Hasse-Weil L-functions affect the distribution of Frobenius traces in function field settings?
• RQ5 Can unconditional results on bias be obtained in function fields where number field analogues require GRH and LI?

Key findings

• In generic families of elliptic curves over function fields, the bias in the race between positive and negative Frobenius traces diminishes as the conductor grows, with the logarithmic density converging to 1/2.
• The central limit theorem for Frobenius traces implies that the distribution of the normalized trace becomes Gaussian, leading to vanishing bias in the limit.
• The proportion of curves with nontrivial relations among the zeros of their L-functions is bounded by O(n q^{-n d^{-2}(14 + b_E/d)^{-1}} log q), which decays rapidly with n.
• In Ulmer’s family, non-generic zero dependencies lead to persistent, extreme biases—some of which are believed to be impossible in the number field case.
• The existence of such strong biases is due to the fact that the zeros of the L-functions in this family are highly dependent, a situation that is non-generic and provably rare in general.
• Unconditional results are obtained in the function field setting, avoiding the need for GRH or LI, which are required in the number field analogues.