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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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[1] Baig, S., Hall, Chr.: Experimental data for Goldfeld’s conjecture over function fields, Exp. Math., 21, 362-374, (2012) · Zbl 1323.11033
[2] Brumer, A., The average rank of elliptic curves I, Invent. Math., 109, 445-472, (1992) · Zbl 0783.14019
[3] Conceição, R.P., Hall, Chr., Ulmer, D.: Explicit points on the Legendre curve II. Math. Res. Lett. 21(2), 261-280 (2014) · Zbl 1312.14065
[4] Fisher, B.: Equidistribution theorems. In: Columbia University Number Theory Seminar (New York, 1992). Astérisque, vol. 228(3), pp. 69-79. Société Mathématique de France, Paris (1995)
[5] Fu, L.; Liu, C., Equidistribution of Gauss sums and Kloosterman sums, Math. Z., 249, 269-281, (2005) · Zbl 1136.11052
[6] Griffon, R.: Analogues du Théorème de Brauer-Siegel pour Quelques Familles de Courbes Elliptiques. PhD thesis, Université Paris Diderot (2016). http://math.leidenuniv.nl/ griffonrmm/thesis/Griffon_thesis.pdf
[7] Griffon, R., Analogue of the Brauer-Siegel theorem for Legendre elliptic curves, J. Number Theory, 193, 189-212, (2018) · Zbl 1440.11093
[8] Griffon, R.: Explicit \(L\)-functions and a Brauer-Siegel theorem for Hessian elliptic curves. J. Théor. Nombres Bordeaux. arXiv:1709.02761 (to appear) · Zbl 1440.11093
[9] Hindry, M.: Why is it difficult to compute the Mordell-Weil group? In: Zannier, U. (ed.) Diophantine Geometry. Centro di Ricerca Matematica Ennio De Giorgi Series, vol. 4, pp. 197-219. Edizioni della Normale, Pisa (2007) · Zbl 1219.11099
[10] Hindry, M.; Pacheco, A., An analogue of the Brauer-Siegel theorem for abelian varieties in positive characteristic, Moscow Math. J., 16, 45-93, (2016) · Zbl 1382.11041
[11] Katz, N.M.: Gauss Sums, Kloosterman Sums, and Monodromy Groups. Annals of Mathematics Studies, vol. 116. Princeton University Press, Princeton (1988) · Zbl 0675.14004
[12] Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Dover, New York (2006) · Zbl 0281.10001
[13] Lang, S.; Artin, M. (ed.); Tate, J. (ed.), Conjectured Diophantine estimates on elliptic curves, No. 35, 155-171, (1983), Boston
[14] Lidl, R., Niederreiter, H.: Finite fields, 2nd edn. In: Encyclopedia of Mathematics and Its Applications, vol. 20. With a foreword by P.M. Cohn. Cambridge University Press (1997)
[15] Milne, J.S.: Arithmetic Duality Theorems. Perspectives in Mathematics, vol. 1. Academic Press, Boston (1986)
[16] Mignotte, M.; Waldschmidt, M., On algebraic numbers of small height: linear forms in one logarithm, J. Number Theory, 47, 43-62, (1994) · Zbl 0801.11033
[17] Niederreiter, H., The distribution of values of Kloosterman sums, Arch. Math. (Basel), 56, 270-277, (1991) · Zbl 0752.11055
[18] Pries, R.; Ulmer, D., Arithmetic of abelian varieties in Artin-Schreier extensions, Trans. Amer. Math. Soc., 368, 8553-8595, (2016) · Zbl 1410.11061
[19] Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 151. Springer, New York (1994)
[20] Tate, J.T.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. In: Séminaire Bourbaki, vol. 9, Exp. No. 306, pp. 415-440. Société Mathématique de France, Paris (1965/1966)
[21] Ulmer, D.: Elliptic curves over function fields. In: Popescu, C. et al. (eds.) Arithmetic of \(L\)-Functions. IAS/Park City Math. Series, vol. 18, pp. 211-280. American Mathematical Society, Providence (2011) · Zbl 1323.11037
[22] Geer, G.; Vlugt, M., Kloosterman sums and the \(p\)-torsion of certain Jacobians, Math. Ann. (Basel), 290, 549-563, (1991) · Zbl 0731.14014
[23] Ulmer, D.: On the Brauer-Siegel ratio for abelian varieties over function fields. arXiv:1806.01961 (preprint)
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