Research
The links in
green are notes from conference proceedings.
On the relative trace formula
- A relative trace formula for a compact Rieman surface, with
Mark McKee and Eric Wambach
Submitted (revised Nov 12, 2009).
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
Amer. J. Math., to appear (revised Jan 28, 2009).
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)
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 following paper, ending with
an outline of the relative trace formula approach to proving special value
formulas.
- Central L-values and toric periods
for GL(2), with David Whitehouse
Int. Math. Res. Not. 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, improving results of Waldspurger.
[Old version (Nov. 13, 2006). This uses a simpler trace formula but is much less general.]
-
Transfer from GL(2,D) to GSp(4)
Proceedings of the 9th Autumn Workshop on Number Theory,
Hakuba, Japan.
These are notes from a talk explaining an application of my work with Jacquet
(below) to the question of transferring representations to GSp(4).
- Shalika periods on GL(2,D) and GL(4),
with Herve Jacquet
[preprint version]
Pacific J. Math., 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).
On Artin L-functions
- Four-dimensional Galois representations of solvable type and automorphic forms
[abstract]
Ph.D. Thesis, Caltech, 2004.
The main results are contained in the two papers below, though the thesis
includes minor additional work.
I also 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.
-
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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