Research
(This page will be updated soon to account for some recent developments in my research.)
(This page will be updated soon to account for some recent developments in my research.)
My research interests are broadly in the interface of Number Theory, Representation Theory, Algebra, and Geometry. I work with objects relevant to the Langlands Program. More specifically, I work with automorphic representations and their associated L-functions. I study the special values of automorphic L-functions using analytic techniques from the Langlands program and geometric techniques involving the cohomology of locally symmetric spaces.
My venture into special values started with my work with Freydoon Shahidi where we defined certain periods for cohomological cuspidal automorphic representations for GL(n) over a number field, building on previous work of Harder, Hida and Mahnkopf. We studied the behavior of these periods under twisting by algebraic Hecke Characters. See our 2008-IMRN paper.
I studied the special values of L-functions for GL(n) x GL(n-1) over Q. This can be used to prove results for special values of odd symmetric power L-functions. See my 2010-IMRN paper; this built on the papers of Kazhdan-Mazur-Schmidt in Crelle 2000 and Mahnkopf in Jussieu 2005. The main point is that standard integral representation for these L-functions can be seen as a pairing arising in the context of cohomology of arithmetic groups. I then generalized this to L-functions for GL(n) x GL(n-1) over any number field in 2016 paper in Forum Math. My student Gunja Sachdeva and I, building on papers of Mahnkopf and his student Geroldinger as well as my own Forum paper, studied the special values of L-functions for GL(3) x GL(1) over a totally real field.
Günter Harder and I are studied Eisenstein Cohomology for GL(N), with N odd, to prove results on ratios of successive critical values for Rankin-Selberg L-functions for GL(a) x GL(b), with a + b = N. See our 2011 Comptes Rendus paper for an announcement of the results for this situation over Q. I was curious if my results with Harder can be viewed from the perspective of Deligne's conjecture on special values of motivic L-functions. In Fall 2012, Chandrasheel Bhagwat and I proved some period relations for tensor product motives which together with Deligne's conjecture exactly predicts the results with Harder. Beautiful! In 2015, Harder and I submitted our 80 page manuscript with all the proofs of the results in our announcement but now working over any totally real number field. I am currently working on generalizing this project to Eisenstein cohomology for GL(N) over a CM field. Separately, Bhagwat and I are studying Eisenstein cohomology for even orthogonal groups giving us rationality results for ratios of critical values for SO(n,n) x GL(1). See the announcement of our results in Comptes Rendus in 2016. The long version of our manuscript is getting written up.
I have studied the special values of the standard L-functions for GL(2n) when the representation at hand has a Shalika model, that is if it is a Langlands transfer from GSpin(2n+1). This is joint work with Harald Grobner; see our 2014 paper in American Journal of Math.This work gives new results on several classical situations: symmetric cube L-functions for Hilbert modular forms, degree four L-functions of Siegel modular forms, etc. Recently, Mladen Dmitrov, Fabian Januszewski and I extended this work to construct p-adic L-functions in this context, and after proving appropriate Manin relations that relate the p-adic L-functions interpolating the values at different critical points, we can prove nonvanishing of twists of the complex L-function at the center of symmetry.
If one is interested in special values of L-functions then one is forced to study the arithmetic properties of automorphic forms. My student Naomi Tanabe and I started out by studying arithmetic properties of Hilbert modular forms. This is of course all very well-known but for us it was an educational exercise exploiting the point of view afforded by my period relations with Shahidi: these relations say that it suffices to prove a theorem for L(1/2, pi), then we automatically have a theorem for every critical value of every twisted L-function L(s, pi x chi). See our 2011-JRMS paper.
Harald Grobner and I studied the structure of automorphic cohomology for inner forms of GL(n). We studied the action of automorphisms of complex numbers on the cohomology of arithmetic subgroups of GL(n,D). This appeared in IJNT in 2014.
Wee Teck Gan and I studied a dictum that we call arithmeticity for periods of automorphic forms. If a cusp form on a group G has a nonzero period relative to a subgroup H then whether every Galois conjugate of this cusp form also has this property. This question is intimately related to the intrinsic structure of automorphic forms on G as well as to the special values of L-functions.
Baskar Balasubramanyam and I studied the special values of adjoint L-functions for GL(n), and proved that if a prime divides the algebraic part of the value at s=1 of the adjoint L-function of a cohomological cuspidal representation of GL(n) over any number field, and supposing this prime is outside a finite set of exceptional primes, then this prime is a congruence prime for the representation. See our paper this year in American Journal of Math.
My first love in Mathematics was the representation theory of p-adic groups. In my PhD thesis I studied the representations of p-adic GL(2,D) for a division algebra over a p-adic field. The thesis was joint work with Dipendra Prasad. We developed a Kirillov theory for such representations. See our 2000-Duke paper. In later work, I used this Kirillov theory to study the restriction of representations of GL(2,D) to the diagonal subgroup D* x D* in GL(2,D). See my 2007-CJM paper. For this CJM paper, I needed to prove a Künneth theorem for p-adic groups which I did using some Bushnell-Kutzko machinery; such homological results are interesting in their own right. See my 2007-CMB paper. In another direction with p-adic groups, Joshua Lansky and I studied to what extent Casselman's classical results on new forms will generalize to other groups. We studied two cases closely related to GL(2): namely, SL(2) and U(1,1). See our 2007-PJM paper and our 2004-PIAS paper.
I have, in the past, dabbled with a bit of analytic number theory starting with a paper with Ram Murty in JRMS in 2000. This paper uses some cute lemmas in representations of finite groups to prove results towards the classical Dedekind's conjecture which states that the only pole for the zeta function of a number field should be coming from the pole of the Riemann zeta function. Later, when I was visiting Ram Murty at Queen's University he asked me if one could prove a nonvanishing result for Rankin-Selberg L-functions L(s, f x g), where f is fixed, and the point s is fixed and the game is to find a twisting g to make the L-value nonzero. Such a result is proved in my 2002 Canadian Comptes Rendus paper.