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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:
NTL; ELLFF; SageMath
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##### References:
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