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Ameya Pitale

Publications and source records attributed to Ameya Pitale.

30 records · Page 2Linked to original sources
arXiv2014

Lifting to GL(2) over a quaternion division algebra and an explicit construction of CAP representations

The aim of this paper is to carry out an explicit construction of CAP representations of GL(2) over a division quaternion algebra with discriminant two. We first construct cusp forms on such group explicitly by lifting from Maass cusp forms for the congruence subgroup of level 2. We show that this lifting is non-zero and Hecke-equivariant. This allows us to determine each local component of such a cuspidal representation. We then know that our cuspidal representations provide examples of CAP representations, and in fact, counterexamples of the Generalized Ramanujan conjecture.

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arXiv2014

Local and global Maass relations (expanded version)

We characterize the irreducible, admissible, spherical representations of GSp(4,F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied by the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito-Kurokawa lifting. We show how the classical Maass relations can be deduced from the local relations in a representation theoretic way, without recourse to the construction of Saito-Kurokawa lifts in terms of Fourier coefficients of half-integral weight modular forms or Jacobi forms. As an additional application of our methods, we give a new characterization of Saito-Kurokawa lifts involving a certain average of Fourier coefficients.

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arXiv2013

Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels

Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk -> infinity. This generalizes known results in the case that q is squarefree. We obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime. We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions of Watson's formula to certain triple product integrals involving forms of non-squarefree level. By a theorem of Ichino and a lemma of Michel--Venkatesh, this amounts to a detailed study of Rankin--Selberg integrals int|f|^2 E attached to newforms f of arbitrary level and Eisenstein series E of full level. We find that the local factors of such integrals participate in many amusing analogies with global L-functions. For instance, we observe that the mass equidistribution conjecture with a power savings in the depth aspect is equivalent to the union of a global subconvexity bound and what we call a "local subconvexity bound"; a consequence of our local calculations is what we call a "local Lindelof hypothesis".

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arXiv2013

Characterizations of the Saito-Kurokawa lifting: a survey

There are a variety of characterizations of Saito-Kurokawa lifts from elliptic modular forms to Siegel modular forms of degree 2. In addition to giving a survey of known characterizations, we apply a recent result of Weissauer to provide a number of new and simpler characterizations of Saito-Kurokawa lifts.

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arXiv2013

Transfer of Siegel cusp forms of degree 2

Let $π$ be the automorphic representation of $\GSp_4(\A)$ generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and $τ$ be an arbitrary cuspidal, automorphic representation of $\GL_2(\A)$. Using Furusawa's integral representation for $\GSp_4\times\GL_2$ combined with a pullback formula involving the unitary group $\GU(3,3)$, we prove that the $L$-functions $L(s,π\timesτ)$ are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations $π$ have a functorial lifting to a cuspidal representation of $\GL_4(\A)$. Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of $π$ to a cuspidal representation of $\GL_5(\A)$. As an application, we obtain analytic properties of various $L$-functions related to full level Siegel cusp forms. We also obtain special value results for $\GSp_4\times\GL_1$ and $\GSp_4\times\GL_2$.

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arXiv2009

Steinberg representation of GSp(4): Bessel models and integral representation of L-functions

We obtain explicit formulas for the test vector in the Bessel model and derive the criteria for existence and uniqueness for Bessel models for the unramified, quadratic twists of the Steinberg representation πof GSp(4,F), where F is a non-archimedean local field of characteristic zero. We also give precise criteria for the Iwahori spherical vector in πto be a test vector. We apply the formulas for the test vector to obtain an integral representation of the local L-function of πtwisted by any irreducible, admissible representation of GL(2,F). Together with results in \cite{Fu} and \cite{PS2}, we derive an integral representation for the global L-function of the irreducible, cuspidal automorphic representation of GSp(4,A) obtained from a Siegel cuspidal Hecke newform, with respect to a Borel congruence subgroup of square-free level, twisted by any irreducible, cuspidal, automorphic representation of GL(2,A). A special value result for this L-function in the spirit of Deligne's conjecture is obtained.

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arXiv2009

Integral representation for L-functions for GSp(4) x GL(2), II

Based on Furusawa's theory, we present an integral representation for the L-function L(s,π\times τ), where πis a cuspidal automorphic representation on GSp(4) related to a holomorphic Siegel modular form, and where τis an arbitrary cuspidal automorphic representation on GL(2). As an application, a special value result for this L-function in the spirit of Deligne's conjecture is proved.

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arXiv2008

Bessel models for lowest weight representations of GSp(4,R)

We prove uniqueness and give precise criteria for existence of split and non-split Bessel models for a class of lowest and highest weight representations of the groups GSp(4,R) and Sp(4,R) including all holomorphic and anti-holomorphic discrete series representations. Explicit formulas for the resulting Bessel functions are obtained by solving systems of differential equations. The formulas are applied to derive an integral representation for a global $L$-function on GSp(4)xGL(2) involving a vector-valued Siegel modular form of degree 2.

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arXiv2008

L-functions for GSp(4)xGL(2)in the case of high GL(2) conductor

Furusawa has given an integral representation for the degree 8 L-function of GSp(4) x GL(2) and has carried out the unramified calculation. The local p-adic zeta integrals were calculated in our earlier work under the assumption that the GSp(4) representation πis unramified and the GL(2) representation τhas conductor p. In the present work we generalize to the case where the GL(2) representation has arbitrarily high conductor. The result is that the zeta integral represents the local Euler factor L(s,π\times τ) in all cases. As a global application we obtain a special value result for a GSp(4) x GL(2) global L-function coming from classical holomorphic cusp forms with arbitrarily high level for the elliptic modular form.

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arXiv2008

Integral Representaion for L-functions for GSp(4)xGL(2)

Let pi be a cuspidal, automorphic representation of GSp(4) attached to a Siegel modular form of degree 2. We refine the method of Furusawa to obtain an integral representation for the degree-8 L-function L(s,pi x tau), where tau runs through certain cuspidal, automorphic representation of GL(2). Our calculations include the case of square-free level for the p-adic components of tau, and a wide class of archimedean types including Maass forms. As an application we obtain a special value result for L(s,pi x tau).

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