Eisenstein series twisted by modular symbols for the group $\SL _{n}$

Abstract

Abstract. We define Eisenstein series twisted by modular symbols for the group SL n , generalizing a construction of the first author [1,2]. We show that, in the case of series attached to the minimal parabolic subgroup, our series converges for all points in a suitable cone. We conclude with examples for SL 2 and SL 3 .

Invariants, cohomology, and automorphic forms of higher order

“…For instance, the limit crossing probabilities in percolation theory, considered as functions of the aspect ratio, turn out to be higher order forms [KZ03]. As an approach to the abcconjecture, Dorian Goldfeld introduced Eisenstein-series twisted by modular symbols [CDO02, GG00,Gol02], which are higher order forms. Finally, spaces of higher order forms are natural receptacles of converse theorems [Far02,IM06].…”

Higher order invariants in the case of compact quotients

“…Higher order modular forms show up in various contexts, for instance in percolation theory [16], or in the theory of Eisenstein-series twisted by modular symbols [2,12,13]. Finally, spaces of higher order forms are natural receptacles of converse theorems [10,15].…”

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

“…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.…”