Arithmetic statistics of modular symbols

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.

Modular forms of virtually real-arithmetic type I: Mixed mock modular forms yield vector-valued modular forms

“…The range of covered topics includes combinatorics, geometry, representation theory, and string theory [3,49]. Higher order modular forms, first introduced with this name by Chinta, Diamantis, and O'Sullivan [18] and Kleban and Zagier [37], have served as a handle on the distribution of modular symbols [52,53] and their connection to the ABC-conjecture [32], and made appearance in conformal field theory [22,37]. Recently, iterated Eichler-Shimura integrals have received a modular interpretation analogous to the one of mock modular forms [10], and at the same time were equipped with a motivic-geometric interpretation [2,11].…”

Additive twists and a conjecture by Mazur, Rubin and Stein

“…Based on both theoretical and computational arguments (the latter jointly with W. Stein) they formulated a number of precise conjectures. We state one of them in its formulation given in [11].…”

“…Then computing the integral using the Fourier expansion of f gives us the right hand side of the above formulas. An average version of this conjecture in the case of square-free levels was proved in [11]. The same paper contains the proofs of other conjectures from the original set listed in [10].…”

Distribution of modular symbols in $\mathbb{H}^3$

“…By the conjectures of Birch-Swinnerton-Dyer, this is related to studying when there is excess rank rankE(L) > rankE(Q) , where L/Q is a cyclic extension. Motivated by this, the study of the distribution of modular symbols became a very active area; see the work of Petridis-Risager [16], [17], [18], Diamantis-Hoffstein-Kıral-Lee [5], Lee-Sun [12], Bettin-Drappeau [1] and Nordentoft [14]. In this work, we investigate the distribution of modular symbols associated to an imaginary quadratic field.…”

“…Theorem 1.3 (Petridis-Risager [18]). There exist explicit constants C f , D f,ab such that (a) (Normal distribution) The values of…”