Aaron Pollack

Aaron Pollack
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Mathematics - Representation Theory (6)
 
Mathematics - Number Theory (6)

Publications Authored By Aaron Pollack

Suppose $F$ is a non-archimedean local field. The classical Godement-Jacquet theory is that one can use Schwartz-Bruhat functions on $n \times n$ matrices $M_n(F)$ to define the local standard $L$-functions on $\mathrm{GL}_n$. The purpose of this partly expository note is to give evidence that there is an analogous and useful "approximate" Godement-Jacquet theory for the standard $L$-functions on the special orthogonal groups $\mathrm{SO}(V)$: One replaces $\mathrm{GL}_n(F)$ with $\mathrm{GSpin}(V)(F)$ and $M_n(F)$ with $\mathrm{Clif}(V)(F)$, the Clifford algebra of $V$. Read More

In this paper we discuss lifting laws. Roughly, lifting laws are ways of "lifting" elements of the open orbit of one prehomogeneous vector space to elements of the minimal nonzero orbit of another prehomogeneous vector space. We prove a handful of these lifting laws, and show how they can be used to help solve certain problems in arithmetic invariant theory. Read More

We give a Rankin-Selberg integral representation for the Spin (degree eight) $L$-function on $\mathrm{PGSp}_6$. The integral applies to the cuspidal automorphic representations associated to Siegel modular forms. If $\pi$ corresponds to a level one Siegel modular form $f$ of even weight, and if $f$ has a non-vanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $\Lambda(\pi,Spin,s)$ of $\pi$. Read More

In this paper we give Rankin-Selberg integrals for the quasisplit unitary group on four variables, $\mathrm{GU}(2,2)$, and a closely-related quasisplit form of $\mathrm{GSpin}_6$. First, we give a two-variable Rankin-Selberg integral on $\mathrm{GU}(2,2)$. This integral applies to generic cusp forms, and represents the product of the exterior square (degree six) $L$-function and the standard (degree eight) $L$-function. Read More

We give two global integrals that unfold to a non-unique model and represent the partial Spin $L$-function on $\mathrm{GSp}_6$. We deduce that for a wide class of cuspidal automorphic representations $\pi$, the partial Spin $L$-function is holomorphic except for a possible simple pole at $s=1$, and that the presence of such a pole indicates that $\pi$ is an exceptional theta lift from $\mathrm{G}_2$. These results utilize and extend previous work of Gan and Gurevich, who introduced one of the global integrals and proved these facts for a special subclass of these $\pi$ upon which the aforementioned model becomes unique. Read More

The Rankin-Selberg integral of Kohnen and Skoruppa produces the Spin $L$-function for holomorphic Siegel modular forms of genus two. In this paper, we reinterpret and extend their integral to apply to arbitrary cuspidal automorphic representations of $\mathrm{PGSp}_4$. We show that the integral is related to a non-unique model and analyze it using the approach of Piatetski-Shapiro and Rallis. Read More