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April 10, 2015 By

Variation in the number of points on elliptic curves and applications to excess rank

C. R. Math. Rep. Acad. Sci. Canada Vol. 27, (4), 2005 pp. 111–120

December 30, 2005

Steven J. Miller, Department of Mathematics, Brown University, 151 Thayer Street, Providence, Rhode Island 02912 USA; email: sjmiller@math.brown.edu

Abstract/Résumé:

Michel proved that for a one-parameter family of elliptic curves over \(\mathbb{Q}(T)\) with non-constant \(j(T)\) that the second moment of the number of solutions modulo \(p\) is \(p^2 + O(p^{3/2})\). We show this bound is sharp by studying \(y^2 = x^3 + Tx^2 + 1\). Lower order terms for such moments in a family are related to lower order terms in the \(n\)-level densities of Katz and Sarnak, which describe the behavior of the zeros near the central point of the associated \(L\)-functions. We conclude by investigating similar families and show how the lower order terms in the second moment may affect the expected bounds for the average rank of families in numerical investigations.

Michel a démontré que pour une famille de courbes élliptiques à un paramètre sur \(\mathbb{Q}(T)\) avec \(j(T)\) non-constant, le second moment du nombre de solutions modulo \(p\) est \(p^2 + O(p^{3/2})\). Nous montrons que cette limite est précise en étudiant \(y^2 = x^3 + Tx^2 + 1\). Pour de tels moments dans une famille, les termes d’ordre inférieur sont liés aux termes dans les \(n\)-niveaux de densité de Katz et Sarnak, qui decrivent le comportement des zéros près du point central des \(L\)-fonctions associées. Nous concluons en recherchant des familles semblables et en montrant comment les termes d’ordre inférieur dans le second moment peuvent affecter les bornes pour le rang moyen de familles dans des simulations numériques.

Keywords: Michel’s theorem, average rank, n-level densities, number of points on elliptic curves
AMS Subject Classification: Elliptic curves over global fields 11G05

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