There are two ways to do great mathematics. The first is to be smarter than everybody else.
The second way is to be stupider than everybody else -- but persistent.
-- Raoul Bott

Links

Research

The links in green are notes from conference proceedings. Please contact me for a copy of any paper without a link.

On special L-values, periods and the relative trace formula

  • On central critical values of the degree four L-functions for GSp(4): a simple trace formula, with Masaaki Furusawa
    Submitted.
    As an application of the Fundamental Lemma I and III papers, we show, for a restricted set of cuspidal automorphic representations of GSp(4), non-vanishing Novodvorsky periods (i.e., twisted L-values) imply non-vanishing Bessel periods for a suitable Jacquet-Langlands transfer, and a converse result. This provides global evidence for Böcherer's conjecture.

  • On central critical values of the degree four L-functions for GSp(4): the fundamental lemma III, with Masaaki Furusawa and Joseph Shalika
    Memoirs of the AMS, to appear.
    We extend the fundamental lemma from our American Journal paper below, as well as one due to Furusawa-Shalika, to the full Hecke algebra.

  • A relative trace formula for a compact Riemann surface, with Mark McKee and Eric Wambach (Errata to published version) [Corrected version (Feb. 2, 2012)]
    International Journal of Number Theory, Vol. 7, No. 2 (2011), pp. 389-429.
    We interpret a relative trace formula on a hyperbolic compact Riemann surface as a relation between the period spectrum and ortholength spectrum of a given closed geodesic. This leads to various asymptotic results on periods and ortholengths, as well as some simultaneous nonvanishing results for two different periods.

  • On central critical values of the degree four L-functions for GSp(4): the fundamental lemma II, with Masaaki Furusawa [preprint version]
    American Journal of Mathematics, Vol. 133, No. 1 (2011), pp. 197-233.
    We propose a different kind of relative trace formula than Furusawa-Shalika to relate central spinor L-values to Bessel periods, and prove the corresponding fundamental lemma. This relative trace formula has several advantages over the previous ones.

  • Central L-values and toric periods for GL(2), with David Whitehouse
    International Mathematical Research Notices (IMRN) 2009, No. 1 (2009), pp. 141-191.
    Using Jacquet's relative trace formula, we get a formula for the central value of a GL(2) L-function, refining results of Waldspurger.
    [Old version (Nov. 13, 2006). This uses a simpler trace formula but is much less general.]

  • Central L-values and toric periods for GL(2)
    Automorphic Represenations, Automorphic Forms, L-functions and Related Topics, Jan. 21-25, 2008, RIMS, Kyoto, Conference Proceedings.
    This is basically an extended introduction to the above paper, ending with an outline of the relative trace formula approach to proving special value formulas.

  • Shalika periods on GL(2,D) and GL(4), with Herve Jacquet [preprint version]
    Pacific Journal of Mathematics, Vol. 233, No. 2 (2007), pp. 341-370.
    Here we use a relative trace formula to study period integrals, which yield results about exterior-square L-functions, and thus about transfer to GSp(4).

  • Transfer from GL(2,D) to GSp(4)
    Proceedings of the 9th Autumn Workshop on Number Theory, Hakuba, Japan (2006).
    These are notes from a talk explaining an application of my work with Jacquet (above) to the question of transferring representations to GSp(4).
On algebraic number theory On Artin L-functions

Undergraduate Research Supervised

Notes on Number Theory and Representation Theory

Notes on Algebraic Combinatorics

Teaching (Spring 2012)