Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field

Abstract

Introduction 1 2. Special values of the adjoint L-function 4 2.1. Rankin-Selberg integrals for GL n × GL n 4 2.2. An integral representation of L(1, Ad 0 , π) 5 2.3. Ramified calculations 7 3. Whittaker models and automorphic cohomology 8 3.1. The basic set-up to study the cohomology of arithmetic groups 8 3.2. Betti-Whittaker periods 9 3.3. Cohomological pairing and the main theorem on adjoint L-values 11 3.4. On the criticality of the adjoint L-function at s = 1 14 4. Discriminant calculations and cohomological congruences 15 4.1. Some linear algebra 15 4.2. Discriminants and congruence modules 16 4.3. The main theorem on congruences 19 5. The non-vanishing property 20 References 25

Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

“…In this paper, we prove a rationality result for the residue of the exterior square L-function at s = 1 and also for the holomorphic value of the symmetric square L-function at s = 1 attached to Π. On the way, we also show a rationality result for the residue of the Rankin-Selberg L-function at s = 1, which is very much in the spirit of our recent joint paper with Harris and Lapid [11], as well as of one of the main results in a recent article of Balasubramanyam-Raghuram [2]…”

“…Guided by the above methods, meanwhile, there is a growing number of results that have been proved about the rationality of special values of certain automorphic L-functions attached to GL n . As a selection of examples, relevant to the present paper, we refer to Raghuram [28,27], Harder-Raghuram [16], Grobner-Harris [10]; Grobner-Raghuram [13], Grobner-Harris-Lapid [11] and Balasubramanyam-Raghuram [2].…”

“…In order to obtain our second main theorem on the symmetric square L-function, we need to prove a version of one of the main results of Grobner-Harris-Lapid [11] and Balasubramanyam-Raghuram [2], which is tailored to our present situation at hand. This is done in §8.…”

Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts

“…As in the case of elliptic modular forms, congruences between automorphic forms on other reductive algebraic groups have numerous arithmetic applications. One can see [1,2,3,4,5,7,8,13,24,25,27,28,36,37] for examples of such applications. As such, there is considerable interest in classifying congruence primes for automorphic forms on these groups.…”

On the $$\mathrm {GL}_n$$ GL n -eigenvariety and a conjecture of Venkatesh

“…Venkatesh and his collaborators [25,39] have carried this out already for the Hodge and l-adic realizations, using the theory of cuspidal cohomology and (assuming the existence of certain Galois representations) the Taylor-Wiles method. Additional evidence is provided by recent work of Balasubramanyam-Raghuram [8].…”