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Bounds on special values of $$L$$-functions of elliptic curves in an Artin-Schreier family. (English) Zbl 07073693
Summary: We study a certain Artin-Schreier family of elliptic curves over the function field $$\mathbb{F}_q(t)$$. We prove an asymptotic estimate on the special values of their $$L$$-function in terms of the degree of their conductor; we show that the special values are, in a sense, ‘asymptotically as large as possible’. We also provide an explicit expression for their $$L$$-function. The proof of the main result uses this expression and a detailed study of the distribution of character sums related to Kloosterman sums. Via the BSD conjecture, the main result translates into an analogue of the Brauer-Siegel theorem for these elliptic curves.

##### MSC:
 11G05 Elliptic curves over global fields 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11M38 Zeta and $$L$$-functions in characteristic $$p$$ 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11L05 Gauss and Kloosterman sums; generalizations 11J20 Inhomogeneous linear forms
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