The modular symbol and continued fractions in higher dimensions

Modular Forms, a Computational Approach

“…Statement (1) of Theorem A.22 is due to Ash and Rudolph [AR79]. Instead of P , they used the larger parallelotope P ′ defined by…”

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

“…We begin by recalling some facts from the theory of modular symbols associated to the minimal parabolic subgroup. These facts are equivalent to results in [13], and are just reformulated in terms of tuples and splittings. Let W be the set of all full tuples of 1-dimensional subspaces.…”

“…Note that the relations in [12,13] imply that this equality holds true in Proof. We begin by recalling some facts from the theory of modular symbols associated to the minimal parabolic subgroup.…”

“…First, the twisted series on the right are well-defined, since if P and w are compatible then so are P and w(i) for each i ≥ 1. We have the following basic relation among modular symbols for the minimal parabolic subgroup, from [12,13]:…”

Hecke Operators and Hilbert Modular Forms

“…For Γ ⊂ SL n (Z), modular symbols provide a concrete method to compute the Hecke eigenvalues in H ν (Y ; M), where ν = n(n+1)/2−1 is the top nonvanishing degree [8,23]. Using modular symbols many people have studied the arithmetic significance of this cohomology group, especially for n = 2 and 3 [3,6,7,12,25,26]; these are the only two values of n for which H ν (Y ; M) can contain cuspidal cohomology classes, in other words cohomology classes coming from cuspidal automorphic forms on GL(n).…”