Research
The links in
green are notes from conference proceedings.
Please contact me for a copy of any paper without a link.
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How often should you clean your room,
with Krishnan (Ravi) Shankar
Submitted.
We introduce and study a combinatorial optimization problem motivated by
the question, "How often should you clean your room?"
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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
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.
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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, 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.
- Non-unique factorization and principalization
in number fields
Proceedings of the AMS, Vol. 139, No. 9 (2011), 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.
- 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).
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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).
- 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).
- 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.
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A Symplectic Case of Artin's Conjecture
[preprint version]
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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