Denominators of Eisenstein cohomology classes for GL2 over imaginary quadratic fields

Abstract

We study the arithmetic of Eisenstein cohomology classes (in the sense of G. Harder) for symmetric spaces associated to GL 2 over imaginary quadratic fields. We prove in many cases a lower bound on their denominator in terms of a special L-value of a Hecke character providing evidence for a conjecture of Harder that the denominator is given by this L-value. We also prove under some additional assumptions that the restriction of the classes to the boundary of the Borel-Serre compactification of the spaces is integral. Such classes are interesting for their use in congruences with cuspidal classes to prove connections between the special L-value and the size of the Selmer group of the Hecke character.

The torsion in symmetric powers on congruence subgroups of Bianchi groups

“…as follows from the expression of , in coordinates (see for example [Ber08,2.4]). Now let u be a primitive vector in Λ 2 which is a vector of maximal weight for the standard parabolic subgroup of SL 2 (C) in V (m) (i.e.…”

An R = T theorem for imaginary quadratic fields

“…In most cases, this normalization of the Hecke L-value is integral, i.e., lies in the integer ring of a finite extension of F p . See [4] Theorem 3 for the exact statement. Put…”

“…However, the Eisenstein cohomology class used in the proof of [5] Theorem 14 is ordinary because by [4] Lemma 9 its T p -eigenvalue (respectively, T p -eigenvalue) is the p-adic unit pφ 1 (p) + φ 2 (p) (respectively, pφ 1 (p) + φ 2 (p)). Therefore, one can prove the statement for the ordinary cuspidal Hecke algebra.…”

“…For example, Proposition 16 and Theorem 28 of [4] and Proposition 9 and Lemma 11 of [5] imply the following: We will from now on assume that we are either in the situation of Theorem 4.2 or 4.4 and fix the characters φ 1 , φ 2 and φ = φ 1 /φ 2 , with corresponding conditions on the set Σ and definitions of K f , T Σ , and J Σ . We also assume from now on that val p (L int (0, φ)) > 0, however our main result (Theorem 6.13) remains vacuously true when L int (0, φ) is a p-adic unit (see Remark 5.11).…”

Analytic, Reidemeister and homological torsion for congruence three–manifolds

“…The constant c(V ) equals −t (2) (V ) where t (2) (V ) is the L 2 -torsion associated to V , see [4] or [27]. For the trivial representation it equals −1/(6π), for the adjoint representation −13/(6π).…”