Irreducibility Criteria for Local and Global Representations

It is proved that certain types of modular cusp forms generate irreducible automorphic representation of the underlying algebraic group. Analogous archimedean and non-archimedean local statements are also given.Comment: 9 page

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

Transfer of Siegel cusp forms of degree 2

Let $\pi$ be the automorphic representation of \GSp_4(\A) generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and $\tau$ 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 $L(s,\pi\times\tau)$ are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations $\pi$ 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 $\pi$ 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.Comment: 99 pages, with a completely re-written introduction; to appear in Memoirs of the AM

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".Comment: 43 pages; various minor corrections (many thanks to the referee) and improvements in clarity and exposition. To appear in JAM