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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.

MSC:

11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11Y16 Number-theoretic algorithms; complexity

Software:

ELLFF; SageMath; NTL

References:

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[2] Goldfeld [Goldfeld 79] D., Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979 pp 108– (1979)
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[9] Shoup [Shoup 09] V., NTL: A Library for Doing Number Theory (2009)
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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.