Pullbacks of Eisenstein series from GU(3,3) and critical L-values for GSp(4) X GL(2)

Let F be a genus two Siegel newform and g a classical newform, both of squarefree levels and of equal weight l. We prove a pullback formula for certain Eisenstein series -- thus generalizing a construction of Shimura -- and use this to derive an explicit integral representation for the degree eight L-function L(s, F X g). This integral representation involves the pullback of a simple Siegel-type Eisenstein series on the unitary group GU(3,3). As an application, we prove a reciprocity law -- predicted by Deligne's conjecture -- for the critical special values L(m, F X g) where m is an integer, 2 <= m <= l/2-1.Comment: 45 pages; Some notational changes made, inaccuracies eliminated and typos fixed in accordance with an anonymous referee's helpful comments. To appear in the Pacific Journal of Mathematic

L-functions for holomorphic forms on GSp(4) x GL(2) and their special values

We provide an explicit integral representation for L-functions of pairs (F,g) where F is a holomorphic genus 2 Siegel newform and g a holomorphic elliptic newform, both of squarefree levels and of equal weights. When F,g have level one, this was earlier known by the work of Furusawa. The extension is not straightforward. Our methods involve precise double-coset and volume computations as well as an explicit formula for the Bessel model for GSp(4) in the Steinberg case; the latter is possibly of independent interest. We apply our integral representation to prove an algebraicity result for a critical special value of L(s, F \times g). This is in the spirit of known results on critical values of triple product L-functions, also of degree 8, though there are significant differences.Comment: 48 pages, typos corrected, some changes in Sections 6 and 7, other minor change

On ratios of Petersson norms for Yoshida lifts

We prove an algebraicity property for a certain ratio of Petersson norms associated to a Siegel cusp form of degree 2 (and arbitrary level) whose adelization generates a weak endoscopic lift. As a preparation for this, we explicate various features of the correspondence between scalar valued Siegel cusp forms of degree n and automorphic representations on GSp_{2n}.Comment: Several minor changes; 34 page

On sup-norms of cusp forms of powerful level

Let f be an L^2-normalized Hecke--Maass cuspidal newform of level N and Laplace eigenvalue \lambda. It is shown that |f|_\infty <<_{\lambda, \epsilon} N^{-1/12 + \epsilon} for any \epsilon>0. The exponent is further improved in the case when N is not divisible by "small squares". Our work extends and generalizes previously known results in the special case of N squarefree.Comment: Final version, to appear in JEMS. Please also note that the results of this paper have been significantly improved in my recent paper arXiv:1509.07489 which uses a fairly different methodolog

Hybrid sup-norm bounds for Maass newforms of powerful level

Let $f$ be an $L^2$-normalized Hecke--Maass cuspidal newform of level $N$, character $\chi$ and Laplace eigenvalue $\lambda$. Let $N_1$ denote the smallest integer such that $N|N_1^2$ and $N_0$ denote the largest integer such that $N_0^2 |N$. Let $M$ denote the conductor of $\chi$ and define $M_1= M/\gcd(M,N_1)$. In this paper, we prove the bound $|f|_\infty$ $\ll_{\epsilon}$ $N_0^{1/6 + \epsilon} N_1^{1/3+\epsilon} M_1^{1/2} \lambda^{5/24+\epsilon}$, which generalizes and strengthens previously known upper bounds for $|f|_\infty$. This is the first time a hybrid bound (i.e., involving both $N$ and $\lambda$) has been established for $|f|_\infty$ in the case of non-squarefree $N$. The only previously known bound in the non-squarefree case was in the N-aspect; it had been shown by the author that $|f|_\infty \ll_{\lambda, \epsilon} N^{5/12+\epsilon}$ provided $M=1$. The present result significantly improves the exponent of $N$ in the above case. If $N$ is a squarefree integer, our bound reduces to $|f|_\infty \ll_\epsilon N^{1/3 + \epsilon}\lambda^{5/24 + \epsilon}$, which was previously proved by Templier. The key new feature of the present work is a systematic use of p-adic representation theoretic techniques and in particular a detailed study of Whittaker newforms and matrix coefficients for $GL_2(F)$ where $F$ is a local field.Comment: Postprint version; to appear in Algebra and Number Theor

Phase Structure of Higher Spin Black Holes

We revisit the study of the phase structure of higher spin black holes carried out in arXiv$:1210.0284$ using the "canonical formalism". In particular we study the low as well as high temperature regimes. We show that the Hawking-Page transition takes place in the low temperature regime. The thermodynamically favoured phase changes from conical surplus to black holes and then again to conical surplus as we increase temperature. We then show that in the high temperature regime the diagonal embedding gives the appropriate description. We also give a map between the parameters of the theory near the IR and UV fixed points. This makes the "good" solutions near one end map to the "bad" solutions near the other end and vice versa.Comment: References added, Conclusions written in better manner, overall exposition improved, version accepted in JHE

Representations of SL_2(R) and nearly holomorphic modular forms

In this semi-expository note, we give a new proof of a structure theorem due to Shimura for nearly holomorphic modular forms on the complex upper half plane. Roughly speaking, the theorem says that the space of all nearly holomorphic modular forms is the direct sum of the subspaces obtained by applying appropriate weight-raising operators on the spaces of holomorphic modular forms and on the one-dimensional space spanned by the weight 2 nearly holomorphic Eisenstein series. While Shimura's proof was classical, ours is representation-theoretic. We deduce the structure theorem from a decomposition for the space of n-finite automorphic forms on SL_2(R). To prove this decomposition, we use the mechanism of category O and a careful analysis of the various possible indecomposable submodules. It is possible to achieve the same end by more direct methods, but we prefer this approach as it generalizes to other groups. This note may be viewed as the toy case of our paper ["Lowest weight modules of Sp_4(R) and nearly holomorphic Siegel modular forms"], where we prove an analogous structure theorem for vector-valued nearly holomorphic Siegel modular forms of degree two.Comment: 13 page