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Experimental data for Goldfeld’s conjecture over function fields. (English) Zbl 1323.11033
D. Goldfeld’s conjecture [Lect. Notes Math. 751, 108–118 (1979; Zbl 0417.14031)] asserts that the average of the analytic rank of a family of quadratic twists \(E_d\) of a fixed elliptic curve \(E\) over \(\mathbb{Q}\) is \(\frac12\), more precisely \[ \lim_{D \to \infty} \frac{1}{\#\{ d: |d| < D\}}\sum_{|d|<D} \text{rank}(E_d)= \frac{1}{2}. \] The article under review focus on the function field analogue of the conjecture, i.e., when \(\mathbb{Q}\) is replaced by a function field of the type \(K = \mathbb{F}_q(t)\). This has the advantage that \(L\)-functions associated to the elliptic curves become polynomials after a change of variable. In this direction the authors provides some explicit computation of such \(L\)-functions \(L(E/K,s)\) associated to elliptic curves over \(K\), which in particular allow the computation of the analytic rank, namely the order of vanishing of the \(L\)-function at \(s=1\).
The strategy follows the usual approach of expressing the \(L\)-function as an Euler product noting that there exists a finite collection of factors which completely determines the \(L\)-function. The idea is that computing Euler factors for a twist and for a pullback (i.e. passing to a field extension) can be done at a cheaper cost once one knows the Euler factors of the original curve. This is particularly important since, for a fixed prime \(q\), every elliptic curve over \(\mathbb{F}_q(t)\) can be written as the combination of a pullback and a twist of a single elliptic curve \(E_0\) over \(K\).
The authors have written a Sage library called ELLFF for calculating \(L\)-functions of elliptic curves over function fields, whose methods and structure are described in this article. The code is used to gather data from families of non-isogenous elliptic curves over function fields coming from the article [R. Miranda and U. Persson, Math. Z. 193, 537–558 (1986; Zbl 0652.14003)]. The experimental data show evidence that the average analytic rank converges to \(\frac12\) for such families.

11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11Y16 Number-theoretic algorithms; complexity
NTL; ELLFF; SageMath
Full Text: DOI arXiv
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