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
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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 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.
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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
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).
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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).
On algebraic number theory
On Artin L-functions
- 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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