arXiv subjects

A. Raghuram

Publications and source records attributed to A. Raghuram.

At least 19 recordsLinked to original sources

arXiv2026

Betti-Whittaker periods under duality: variations and applications

One of the authors (Chen) had previously proved a result on the behavior of Betti-Whittaker periods under duality for cohomological cuspidal automorphic representations of ${\rm GL}_n/{\mathbb Q}$ under some regularity assumptions while using their relation to $L$-values as an anchor in his proof. In this article we prove a generalization of this result to ${\rm GL}_n$ over any number field $F$ without any regularity assumptions and without recourse to $L$-values, while using the outer-automorphism of ${\rm GL}_n$ as the main tool. Then, using results of Harder and one of the other authors (Raghuram), we give applications to new rationality results for the ratios of special values of general triple product $L$-functions and for general twisted Asai $L$-functions. We also give a new proof of a previous result of Bhagwat and Raghuram on the special values of $L$-functions for orthogonal groups. We present variations on period relations for the Betti-Shalika periods under duality, and the behavior of Betti-Whittaker periods under Galois automorphisms of $F$.

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arXiv2025

Congruences for the ratios of Rankin--Selberg $L$-functions

A well-known principle states that a congruence between objects should give rise to a corresponding congruence between the special values of $L$-functions attached to these objects. We computationally investigate this principle for Rankin--Selberg $L$-functions attached to pairs of holomorphic cuspforms, and formulate a precise conjecture in general.

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arXiv2025

Eisenstein cohomology and congruences for the ratios of Rankin--Selberg $L$-functions

A well-known principle states that a congruence between objects should give rise to a corresponding congruence between the special values of $L$-functions attached to these objects. In this article, using the machinery of Eisenstein cohomology after refining it for integral cohomology, we prove an instance of this principle for the ratios of critical values for Rankin--Selberg $L$-functions attached to pairs of holomorphic cuspforms.

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arXiv2024

Cuspidal cohomology for $GL(n)$ over a number field

The main result of this article proves the nonvanishing of cuspidal cohomology for $GL(n)$ over a number field which is Galois over its maximal totally real subfield. The proof uses the internal structure of a strongly-pure weight that can possibly support cuspidal cohomology and the foundational work of Borel, Labesse, and Schwermer.

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arXiv2024

Ramanujan's congruence primes

Ramanujan showed that $τ(p) \equiv p^{11}+1 \pmod{691}$, where $τ(n)$ is the $n$-th Fourier coefficient of the unique normalized cusp form of weight $12$ and full level, and the prime $691$ appears in the numerator of $ζ(12)/π^{12}$ for the Riemann zeta function $ζ(s)$. Searching for such congruences, it is shown that the prime $67$ appears in the numerator of $L(6,χ)/(π^6 \sqrt{5})$, where $χ$ is the unique nontrivial quadratic Dirichlet character modulo $5$ and $L(s,χ)$ its Dirichlet $L$-function, giving rise to a congruence $f_χ\equiv E^\circ_{6, χ} \pmod{67}$ between a cusp form $f_χ$ and an Eisenstein series $E^\circ_{6, χ}$ of weight $6$ on $Γ_0(5)$ with nebentypus character $χ.$

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arXiv2024

Motives, Periods, and Functoriality

Given a pure motive $M$ over $\mathbb{Q}$ with a multilinear algebraic structure $\mathsf{s}$ on $M$, and given a representation $V$ of the group respecting $\mathsf{s}$, we describe a functorial transfer $M^V$. We formulate a criterion that guarantees when the two periods of $M^V$ are equal. This has an implication for the critical values of the $L$-function attached to $M^V.$ The criterion is explicated in a variety of examples such as: tensor product motives and Rankin-Selberg $L$-functions; orthogonal motives and the standard $L$-function for even orthogonal groups; twisted tensor motives and Asai $L$-functions.

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arXiv2024

On counting totally imaginary number fields

A number field is said to be a CM-number field if it is a totally imaginary quadratic extension of a totally real number field. We define a totally imaginary number field to be of CM-type if it contains a CM-subfield, and of TR-type if it does not contain a CM-subfield. For quartic totally imaginary number fields when ordered by discriminant, we show that about 69.95% are of TR-type and about 33.05% are of CM-type. For a sextic totally imaginary number field we classify its type in terms of its Galois group and possibly some additional information about the location of complex conjugation in the Galois group.

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arXiv2022

Eisenstein Cohomology for GL(N) and the special values of Rankin-Selberg L-functions over a totally imaginary number field

Rationality results are proved for the ratios of critical values of Rankin-Selberg L-functions of GL(n) x GL(n') over a totally imaginary field F, by studying rank-one Eisenstein cohomology for the group GL(N)/F, where N = n+n', generalizing the methods and results of previous work with Guenter Harder where the base field was totally real. In contrast to the totally real situation, the internal structure of the totally imaginary base field has a delicate effect on the rationality results.

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arXiv2022

Notes on the arithmetic of Hecke L-functions

This is an expository article that concerns the various related notions of algebraic idele class characters, the Groessencharaktere of Hecke, and cohomological automorphic representations of GL(1), all under the general title of algebraic Hecke characters. The first part of the article systematically lays the foundations of algebraic Hecke characters. The only pre-requisites are: basic algebraic number theory, familiarity with the adelic language, and basic sheaf theory. Observations that play a crucial role in the arithmetic of automorphic L-functions are also discussed. The second part of the article, on the ratios of successive critical values of the Hecke L-function attached to an algebraic Hecke character, concerns certain variations on a theorem of Guenter Harder, especially drawing attention to a delicate signature that apparently has not been noticed before.

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arXiv2021

Eisenstein cohomology for orthogonal groups and the special values of $L$-functions for ${\rm GL}_1 \times {\rm O}(2n)$

For an even positive integer $n$, we study rank-one Eisenstein cohomology of the split orthogonal group ${\rm O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical values of degree-$2n$ Langlands $L$-functions associated to the group ${\rm GL}_1 \times {\rm O}(2n)$ over $F$. The case $n=2$ specializes to classical results of Shimura on the special values of Rankin - Selberg $L$-functions attached to a pair of Hilbert modular forms.

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arXiv2021

An arithmetic property of intertwining operators for p-adic groups

If one proposes to use the theory of Eisenstein cohomology to prove algebraicity results for the special values of automorphic L-functions as in my work with Harder for Rankin-Selberg L-functions, or its generalizations as in my work with Bhagwat for L-functions for orthogonal groups and independently with Krishnamurthy on Asai L-functions, then in a key step, one needs to prove that the normalised standard intertwining operator between induced representations for p-adic groups has a certain arithmetic property. The principal aim of this article is to address this particular local problem in the generality of the Langlands-Shahidi machinery. The main result of this article is invoked in some of the works mentioned above, and I expect that it will be useful in future investigations on the arithmetic properties of automorphic L-functions.

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arXiv2020

$L$-functions of ${\mathrm{GL}}(2n):$ $p$-adic properties and non-vanishing of twists

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $Π$ of $\mathrm{GL}(2n)$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive since we draw heavily upon the methods used in the recent and separate works of all the three authors. By construction our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover we work under a weaker Panchishkine type condition on $Π_p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$-adic $L$-functions at all critical points. This has the striking consequence that, given a unitary $Π$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $Π_p$ is ordinary, the central critical value $L(\tfrac12, Π\otimesχ)$ is non-zero for all except finitely many Dirichlet characters $χ$ of $p$-power conductor.

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arXiv2019

On the arithmetic of Shalika models and the critical values of $L$-functions for ${\rm GL}(2n)$

Let $Π$ be a cohomological cuspidal automorphic representation of ${\rm GL}_{2n}(\mathbb A)$ over a totally real number field $F$. Suppose that $Π$ has a Shalika model. We define a rational structure on the Shalika model of $Π_f$. Comparing it with a rational structure on a realization of $Π_f$ in cuspidal cohomology in top-degree, we define certain periods $ω^ε(Π_f)$. We describe the behaviour of such top-degree periods upon twisting $Π$ by algebraic Hecke characters $χ$ of $F$. Then we prove an algebraicity result for all the critical values of the standard $L$-functions $L(s, Π\otimes χ)$; here we use the work of B. Sun on the non-vanishing of a certain quantity attached to $Π_\infty$. As an application, we obtain new algebraicity results in the following cases: Firstly, for the symmetric cube $L$-functions attached to holomorphic Hilbert modular cusp forms; we also discuss the situation for higher symmetric powers. Secondly, for Rankin-Selberg $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$; assuming Langlands Functoriality, this generalizes to Rankin-Selberg $L$-functions of ${\rm GL}_n \times {\rm GL}_{n-1}$. Thirdly, for the degree four $L$-functions for ${\rm GSp}_4$. Moreover, we compare our top-degree periods with periods defined by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.

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arXiv2016

Special Values of L-functions for Orthogonal Groups

This is an announcement of certain rationality results for the critical values of the degree-2n L-functions attached to GL(1) $\times$ SO(n, n) over $\mathbb Q$ for an even positive integer n. The proof follows from studying the rank-one Eisenstein cohomology for SO(n + 1, n + 1).

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arXiv2015

Eisenstein Cohomology for GL(N) and ratios of critical values of Rankin-Selberg L-functions

The aim of this article is to study rank-one Eisenstein cohomology for the group GL(N)/F, where F is a totally real field extension of Q. This is then used to prove rationality results for ratios of successive critical values for Rankin-Selberg L-functions for GL(n) x GL(n') over F with the parity condition that nn' is even. The key idea is to interpret Langlands's constant term theorem in terms of Eisenstein cohomology.

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arXiv2015

Endoscopy and the cohomology of GL(n)

In this article we study the nonvanishing of cuspidal cohomology for GL(n). Using endoscopic transfer from various classical groups we construct cuspidal representations of GL(n) of cohomological type while working over a totally real field or a totally imaginary quadratic extension of a totally real field. Generalizing a construction of Laurent Clozel, we also prove nonvanishing of cuspidal cohomology of GL(2n) over any number field but only for coefficient systems coming from parallel weights. Working at an arithmetic level, we also draw some inferences on an endoscopic stratification of inner cohomology of GL(n).

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arXiv2015

Critical values of Rankin-Selberg L-functions for GL(n) x GL(n-1) and the symmetric cube L-functions for GL(2)

In a previous article we had proved an algebraicity result for the central critical value for L-functions for GL(n) x GL(n-1) over Q assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this article is to generalize that result for all critical values for L-functions for GL(n) x GL(n-1) over any number field F. Binyong Sun has recently proved that nonvanishing hypothesis and so the results of this article are unconditional. Using such results for the case of GL(3) x GL(2), new unconditional algebraicity results for the special values of symmetric cube L-functions for GL(2) over F have been proved.

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arXiv2015

p-adic L-functions for GL(n)

These are the expanded notes of a mini-course of four lectures by the same title given in the workshop "p-adic aspects of modular forms" held at IISER Pune, in June, 2014. We give a brief introduction of p-adic L-functions attached to certain types of automorphic forms on GL(n) with the specific aim of understanding the p-adic symmetric cube L-function attached to a cusp form on GL(2) over rational numbers.

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