Modular Symbols have a Normal Distribution

Abstract

Abstract. We prove that the modular symbols appropriately normalized and ordered have a Gaussian distribution for all cofinite subgroups of SL 2 (R). We use spectral deformations to study the poles and the residues of Eisenstein series twisted by power of modular symbols.

A converse theorem for second-order modular forms of level N

“…Then F (τ ) and G(τ ) satisfy the hypothesis of Lemma 1 and by Proposition 3 we get the first part of (B), i.e. the function Λ N (F ; s) +A 0 /s+i k B 0 /(k − s) has a holomorphic continuation to the whole s-plane, is bounded on any vertical strip and satisfies (13). Now let χ be a character as in the theorem.…”

“…It was introduced by Goldfeld in [6] to study the distribution of modular symbols and to provide a new approach to Szpiro's conjecture. Since then, these series have been studied and generalized by many authors ( [2], [4], [7], [8], [11]- [13]). The twisted Eisenstein series is not an automorphic form in the classical sense but satisfies a shifted automorphy relation which involves the ordinary Eisenstein series.…”

Hyperbolic lattice-point counting and modular symbols

“…See [4,5] for their arithmetic significance in the cofinite case of congruence subgroups. In previous work the authors [14,15,18] have studied the distribution of the normalized values of the Poincaré pairing for compact and finite volume hyperbolic surfaces. In all articles we found as limiting distribution the normal Gaussian distribution.…”

The distribution of values of the Poincaré pairing for hyperbolic Riemann surfaces

“…The proof uses the method of asymptotical moments precisely as in [14,17]. From Theorem 4.2, Theorem 3.3 and Lemma 2.1 we may conclude, using a more or less standard contour integration argument (see [14,17] for details), that as T → ∞ (5.1)…”

“…In both articles we found as limiting distribution the normal Gaussian distribution. However, the ordering of the group elements was not geometric: in [14] we ordered the group elements by realizing Γ = π 1 (X) as a discrete subgroup of SL 2 (R), setting γ = a b c d and ordering γ according to c 2 + d 2 . In [17] the matrix elements are ordered according to (a 2 + b 2 )(c 2 + d 2 ).…”