Abstract
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.
Highlights
Modular symbols are fundamental tools in number theory
In this paper we study the arithmetical properties of the modular symbol map a
Remark 1.14 An important tool in this paper and in [36] is non-holomorphic Eisenstein series twisted with modular symbols Em,n(z, s)
Summary
Modular symbols are fundamental tools in number theory. By the work of Birch, Manin, Cremona and others they can be used to compute modular forms, the homology of modular curves, and to gain information about elliptic curves and special values of L-functions. Remark 1.12 We have obtained different but related normal distribution results for modular symbols in [36,37,38,40] One difference between these papers and the current one is in the ordering and normalization of the values of γ , α. Remark 1.14 An important tool in this paper and in [36] is non-holomorphic Eisenstein series twisted with modular symbols Em,n(z, s). These were introduced by Goldfeld [15,16] and studied extensively by many authors, see e.g.
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