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A. Raghuram

Publications and source records attributed to A. Raghuram.

29 records · Page 2Linked to original sources
arXiv2014

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

We prove an integrality result for the value at s=1 of the adjoint L-function associated to a cohomological cuspidal automorphic representation on GL(n) over any number field. We then show that primes (outside an exceptional set) dividing this special value give rise to congruences between automorphic forms. We also prove a non-vanishing property at infinity for the relevant Rankin-Selberg L-functions on GL(n) x GL(n).

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arXiv2013

On some arithmetic properties of automorphic forms of GL(m) over a division algebra

In this paper we investigate arithmetic properties of automorphic forms on the group G' = GL_m/D, for a central division-algebra D over an arbitrary number field F. The results of this article are generalizations of results in the split case, i.e., D=F, by Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic forms and also by Franke and Franke-Schwermer for general automorphic representations. We also compare our theorems on automorphic forms of the group G' to statements on automorphic forms of its split form using the global Jacquet-Langlands correspondence developed by Badulescu and Badulescu-Renard. Beside that we prove that the local version of the Jacquet-Langlands transfer at an archimedean place preserves the property of being cohomological.

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arXiv2013

Ratios of periods for tensor product motives

In this article we prove some period relations for the ratio of Deligne's periods for certain tensor product motives. These period relations give a motivic interpretation for certain algebraicity results for ratios of successive critical values for Rankin-Selberg L-functions for ${\rm GL}_n \times {\rm GL}_{n'}$ proved by Günter Harder and the second author.

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arXiv2013

Comparison results for certain periods of cusp forms on GL(2n) over a totally real number field

This article grew out of my talk in "The Legacy of Srinivasa Ramanujan" conference where I spoke about some techniques to prove algebraicity results for the special values of symmetric cube L-functions attached to the Ramanujan Δ-function. If one wishes to compare these different techniques, then one needs to compare various automorphic periods attached to the symmetric cube transfer of Δ. Motivated by this problem, in this article we provide comparison results for Whittaker-Betti periods, Shalika-Betti periods and certain relative periods attached to a given cohomological cuspidal automorphic representation of GL(2n) over a totally real number field.

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arXiv2012

Arithmeticity for periods of automorphic forms

A cuspidal automorphic representation πof a group G is said to to be distinguished with respect to a subgroup H if the integral of f along H is nonzero for a cusp form f in the space of π. Such period integrals are related to (non)vanishing of interesting L-values and also to Langlands functoriality. This article discusses a general principle, labelled arithmeticity, which roughly states that "πis H-distinguished if and only if any Galois conjugate of πis H-distinguished." We study this principle via several examples; starting with GL(2) and leading up to more complicated situations where the ambient group is a higher GL(n) or a classical group.

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arXiv2011

Notes on the arithmetic of Hilbert modular forms

The purpose of this semi-expository article is to give another proof of a classical theorem of Shimura on the critical values of the standard L-function attached to a Hilbert modular form. Our proof is along the lines of previous work of Harder and Hida (independently). What is different is an organizational principle based on the period relations proved by Raghuram and Shahidi for periods attached to regular algebraic cuspidal automorphic representations. The point of view taken in this article is that one need only prove an algebraicity theorem for the most interesting L-value, namely, the central critical value of the L-function of a sufficiently general type of a cuspidal automorphic representation. The period relations mentioned above then gives us a result for all critical values. To transcribe such a result into a more classical context we also discuss the arithmetic properties of the dictionary between holomorphic Hilbert modular forms and automorphic representations of GL(2) over a totally real number field F.

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arXiv2008

On the special values of certain Rankin-Selberg L-functions and applications to odd symmetric power L-functions of modular forms

We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and Kasten-Schmidt. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands functoriality one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blasius and Panchishkin on twisted L-values in some cases. We comment on the compatibility of our results with Deligne's conjecture on the critical values of motivic L-functions. These results, as in the above mentioned works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals.

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arXiv2007

On certain period relations for cusp forms on GL_n

Let $π$ be a regular algebraic cuspidal automorphic representation of ${\rm GL}_n({\mathbb A}_F)$ for a number field $F$. We consider certain periods attached to $π$. These periods were originally defined by Harder when $n=2$, and later by Mahnkopf when $F = {\mathbb Q}$. In the first part of the paper we analyze the behaviour of these periods upon twisting $π$ by algebraic Hecke characters. In the latter part of the paper we consider Shimura's periods associated to a modular form. If $ϕ_χ$ is the cusp form associated to a character $χ$ of a quadratic extension, then we relate the periods of $ϕ_{χ^n}$ to those of $ϕ_χ$, and as a consequence give another proof of Deligne's conjecture on the critical values of symmetric power $L$-functions associated to dihedral modular forms. Finally, we make some remarks on the symmetric fourth power $L$-functions.

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arXiv2007

Functoriality and special values of L-functions

This is a semi-expository article concerning Langlands functoriality and Deligne's conjecture on the special values of $L$-functions. The emphasis is on symmetric power $L$-functions associated to a holomorphic cusp form, while appealing to a recent work of Mahnkopf on the special values of automorphic $L$-functions.

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