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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.

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
Full Text: DOI
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