arXiv subjects

Freydoon Shahidi

Publications and source records attributed to Freydoon Shahidi.

29 records · Page 2Linked to original sources

arXiv2009

The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

We continue our study of the poles of local Langlands L-functions through the theory of induced from supercuspidal representations of quasi-split groups. Here we study the odd special orthogonal groups, and hence determine poles of Rankin product L-functions. The pole of the intertwining operator is determined in terms of the theory of orbital integrals. This gives a description of the poles in terms of twisted endoscopy, as in previous cases. We use the language of functorial transfer to give precise descrption of the pole in terms of the local components of the global transfer, which has now been established.

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arXiv2007

Residues of Intertwining Operators for Classical Groups with an Appendix "$L$-Functions and Poles of Intertwining Operators"

Let $\tilde{G}$ be a symplectic or even orthogonal group over a p-adic field $F$, and $M$ the Levi factor of a maximal parabolic subgroup of $\tilde{G}$. Suppose that $M$ has the shape of three blocks of the same size. Let $π$ be a supercuspidal representation of $M$. In this paper we give a simple explicit expression for the residue of the standard intertwining operator for the parabolic induction of $π$ from $M$ to $G$.

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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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arXiv2006

Siegel modular forms of genus 2 attached to elliptic curves

The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F) is divisible by an abelian L-function, nor of Yoshida type, in which case L(s,F) is a product of L-series of a pair of elliptic cusp forms. Two key examples are the forms F defined by the symmetric cube of a non-CM elliptic curve E over the rationals, and those defined by anticyclotomic twists (of even non-zero weight) of the base change of such an E to an imaginary quadratic field K. One of the main ingredients is the transfer, albeit indirect, of cusp forms πon GL(4) to GSp(4) with many of the expected properties, and this might be of independent interest. The transfer information is shown to be complete when πis regular and associated to an \ell-adic Galois representation. We also appeal to the (independent) results of Laumon and Weissauer on the zeta functions of Siegel modular threefolds.

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arXiv2005

Generic transfer from GSp(4) to GL(4)

We establish Langlands functoriality for the generic spectrum of GSp(4) and describe its transfer on GL(4). We apply this to prove results toward the generalized Ramanujan conjecture for generic representations of GSp(4).

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arXiv2004

Generic Transfer for General Spin Groups

We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group will not be self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp(4) to GL(4) follows from our result as a special case.

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arXiv2004

Functorial products for $GL_2\times GL_3$ and the symmetric cube for $GL_2$

In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on $GL_2\times GL_3$ as automorphic forms on $GL_6$, from which we obtain our second case, the long awaited functorial symmetric cube map for cusp forms on $GL_2$. We prove these by applying a recent version of converse theorems of Cogdell and Piatetski-Shapiro to analytic properties of certain $L$-functions obtained from the method of Eisenstein series (Langlands-Shahidi method). As a consequence, we prove the bound 5/34 for Hecke eigenvalues of Maass forms over any number field and at every place, finite or infinite, breaking the crucial bound 1/6 (see below and Section 7 and 8) towards Ramanujan-Petersson and Selberg conjectures for $GL_2$. Many other applications are obtained.

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arXiv2003

Automorphic L-functions and functoriality

This is a report on the global aspects of the Langlands-Shahidi method which in conjunction with converse theorems of Cogdell and Piatetski-Shapiro has recently been instrumental in establishing a significant number of new and surprising cases of Langlands Functoriality Conjecture over number fields. They have led to striking new estimates towards Ramanujan and Selberg conjectures.

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arXiv1999

Symmetric cube L-functions for GL_2 are entire

The purpose of this paper is to prove the long awaited holomorphy of the third symmetric power L-functions attached to nonmonomial cusp forms of GL_2 over an arbitrary number field on the whole complex plane.

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arXiv1994

Automorphic $L$\snug-functions, intertwining operators, and the irreducible tempered representations of $p$\snug-adic groups

We give an expository account of the theory of intertwining operators for connected reductive $p$--adic groups, and their connection with automorphic $L$--functions. Our purpose is to illustrate the relation between harmonic analysis and arithmetic. In particular, we describe how the theory of Plancherel measures allows us to compute certain local Langlands $L$--functions. In order to be more self contained, we give a brief introduction to the Langlands program. The theory of $R$--groups and elliptic representations is treated here as well. Finally, we give an example which illustrates how the theory of twisted endoscopy plays a crucial role in determining the poles of both the intertwining operators and the local Langlands $L$--functions in question. A version of this manuscript, in revised form, will appear as a chapter in the proceedings of the conference ``Teoria de Representaciones de Groupos Algebraicos Sobre Cuerpos Locales y Applicationes'' (C. Bushnell, P. Kutzko, J. Pantoja, J. Soto Andrade eds.).

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