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The links in green are notes from conference proceedings. Please contact me for a copy of any paper you cannot download.

  1. Distinguishing graphs with zeta functions and generalized spectra, with Christina Durfee
    Submitted. (Revised: Oct 17, 2014)
    A fundamental problem in graph theory is: when is a graph determined by its spectrum? We investigate analogues of this question with zeta functions in place of spectrum. Our work suggests that zeta functions are more effective at distinguishing graphs than the usual types of spectra studied.

  2. Test vectors and central L-values for GL(2), with Daniel File and Ameya Pitale
    Submitted.
    We extend work of Gross and Prasad on test vectors for GL(2) to cases of joint ramification, and use this to generalize the L-value formula of my IMRN paper with Whitehouse, an average-value formula of Feigon-Whitehouse, and a nonvanishing mod p result of Michel-Ramakrishnan.

  3. Periods and nonvanishing of central L-values for GL(2n), with Brooke Feigon and David Whitehouse
    Submitted. (Revised: Apr 16, 2014)
    Under some local hypotheses, we prove a relation between the nonvanishing of twisted central L-values for GL(2n) and periods over GL(n, E), where E is a quadratic extension. We also deduce analogous local results for supercuspidal representations.

  4. How often should you clean your room? with Krishnan (Ravi) Shankar
    Submitted. (Revised: Aug 31, 2014)
    We introduce and study a combinatorial optimization problem motivated by the question, "How often should you clean your room?"

  5. Local root numbers, Bessel models, and a conjecture of Guo and Jacquet, with Masaaki Furusawa
    Journal of Number Theory, Special Issue in Honor of Steve Rallis, Vol 146 (2015), pp. 150-170.
    We make a conjecture about the transfer of global SO(2)-Bessel periods on SO(2n+1) to GL(n, E) periods on GL(2n), where E is the quadratic extension associated to the relevant form of SO(2), and prove this when n = 2.

  6. On central critical values of the degree four L-functions for GSp(4): a simple trace formula, with Masaaki Furusawa
    Mathematische Zeitschrift, Vol 277, No. 1 (2014), pp. 149-180.
    As an application of the Fundamental Lemma I and III papers, we prove a global Bessel identity for cuspidal automorphic representations of GSp(4) which are supercuspidal at some component (plus some other local hypotheses). In particular, one obtains the global Gross-Prasad Conjecture (a nonvanishing theorem) for such representations.

  7. On central critical values of the degree four L-functions for GSp(4): the fundamental lemma III, with Masaaki Furusawa and Joseph Shalika [preprint version]
    Memoirs of the AMS, Vol 225, No. 1057 (2013), pp. 1-152.
    We extend the fundamental lemma from our American Journal paper below, as well as one due to Furusawa-Shalika, to the full Hecke algebra.

  8. Non-unique factorization and principalization in number fields
    Proceedings of the AMS, Vol. 139, No. 9 (2011), pp. 3025-3038.
    We describe the number and structure of irreducible factorizations of an algebraic integer in the ring of integers of a number field, using what were essentially Kummer's ideas.

  9. 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.

  10. 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.

  11. 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.]

  12. 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.

  13. 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).

  14. 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).

  15. Four-dimensional Galois representations of solvable type and automorphic forms [abstract]
    Ph.D. Thesis, Caltech, 2004.
    This contains the results in the two papers below, as well as a classification of representations into GSp(4,C) of solvable type and minor additional modularity results. I wrote an informal note about my thesis for the layman (by which I mean the mathematically- or scientifically- minded layman).

  16. Modularity of hypertetrahedral representations
    Comptes Rendus Mathematique, Vol. 339, No. 2 (2004), 99-102.
    This proves a new case of modularity for four-dimensional Galois representations induced from a non-normal quartic extension. In particular, one obtains examples of modular representations which are not essentially self-dual.

  17. A symplectic case of Artin's conjecture
    Mathematical Research Letters, Vol. 10, No. 4 (2003), 483-492.
    This gives a new case of Artin's conjecture in GSp(4,C) by establishing the more general Langlands' reciprocity law in this case.

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  • I am on sabbatical for 2014-2015.

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