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My primary interests lie at the intersection of
number theory and algebra, particularly understanding and discovering
algebraic structures in arithmetic, and connecting different types
of structures (modular forms, automorphic representations, representations of
-adic groups, L
-functions, quaternion algebras,
algebraic groups, elliptic curves, ...).
Much of this is done with group theory,
representation theory, and/or harmonic analysis.
I've also done some things with graph theory, combinatorial
optimization and spectral geometry.
Links in green are from conference proceedings.
Please contact me for a copy of any paper you cannot
Rank bias for elliptic curves mod p
with Thomas Pharis
We conjecture that for a fixed prime p, elliptic curves with higher ranks
tend to have more points mod p, and we prove an analogous statement for
Exact double averages of twisted L-values
We prove simple formulas for a double average of twisted central L-values
over both modular forms and twisting characters. This leads to generalizations
of stable (single) average formulas by Michel-Ramakrishnan and Feigon-Whitehouse.
Mass formulas and Eisenstein congruences in higher rank
with Satoshi Wakatsuki
We generalize the approach to constructing Eisenstein congruences
from my 2017 MRL paper to groups of higher rank, with a focus on unitary groups,
though we also give some new results and conjectures about elliptic modular
The relevant Eisenstein series here are associated to minimal parabolic
Refined Goldbach conjectures with primes in progressions
Experimental Mathematics, to appear
We present some refinements of Goldbach's conjectures by restricting
to primes in arithmetic progressions.
An on-average Maeda-type conjecture in the level aspect
Proceedings of the AMS, Vol. 149, No. 4 (2021), 1373-1386
We present a conjecture about the average number of Galois orbits
of newforms for fixed weight and varying level. This has implications
about average ranks of L-functions.
Zeroes of quaternionic modular forms and central L-values
with Jordan Wiebe
Journal of Number Theory, Vol. 217 (2020), 460-494.
We study, theoretically and computationally, zeroes of
modular forms on definite quaternion algebras, and the relation to
non/vanishing of L-values.
The basis problem revisited
Transactions of the AMS, Vol. 373, No. 7 (2020), 4523-4559.
We explicitly describe the Jacquet-Langlands correspondence at the level
of modular forms. This gives a simpler and more flexible solution
to Eichler's basis problem for general level than earlier work of
Hijikata-Pizer-Shemanske for elliptic modular forms, and solves the
basis problem for Hilbert modular forms.
Rationality of Darmon points over genus fields of non-maximal orders
with Matteo Longo and Yan Hu
Annales mathématiques du Québec, Vol. 44, No. 1 (2020), 173-195.
We extend work of Bertolini-Darmon---proving rationality of twists of
Stark-Heegner points, aka Darmon points, for elliptic curves---from
the case of genus characters to quadratic ring class characters.
This uses my 2009 IMRN paper with Whitehouse.
Congruences for modular forms mod 2 and quaternionic S-ideal classes
Canadian Journal of Mathematics, Vol. 70, No. 5 (2018), 1076-1095.
We use quaternionic modular forms to prove various congruences mod 2
between modular forms with differing Atkin-Lehner eigenvalues. The proofs
are related to the distribution of Atkin-Lehner signs (making use of my
"Refined dimensions..." paper below) and
the notion of quaternionic S-ideal classes.
Refined dimensions of cusp forms, and equidistribution and bias of signs
Journal of Number Theory, Vol. 188 (2018), 1-17.
We give dimensions of new spaces of squarefree level with prescribed
Atkin-Lehner eigenvalues or global root numbers, and find these signs are
equidistributed with a strict bias in the weight but perfectly equidistributed
in the level. This is used in my paper above on congruences mod 2.
Periods and nonvanishing of central L-values for GL(2n),
with Brooke Feigon and David Whitehouse
Israel Journal of Mathematics, Vol. 225, No. 1 (2018), 223-266.
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.
- The Jacquet-Langlands
correspondence, Eisenstein congruences, and integral L-values in
weight 2 [errata, corrected version]
Mathematical Research Letters,
Vol. 24, No. 6 (2017), 1775-1795.
We use the Jacquet-Langlands correspondence to generalize
congruence results of Mazur to non-prime level and to
Hilbert modular forms.
finite group characters and refined local-global phenomena,
with Nahid Walji
Acta Arithmetica, Vol. 179, No. 3 (2017), 277-300.
We study the question of how often two finite group characters can agree,
and use this to say how many Euler factors of distinct primitive Artin L-functions
can agree in degree 2 or 3.
Test vectors and central L-values for GL(2),
with Daniel File and Ameya Pitale
Algebra and Number Theory, Vol. 11, No. 2 (2017), 253-318.
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
2009 IMRN paper with Whitehouse, an average-value formula of
Feigon-Whitehouse, and a nonvanishing mod p result of
A comparison of automorphic and
Artin L-series of GL(2)-type agreeing at degree one primes,
with Dinakar Ramakrishnan
Contemporary Mathematics 664, Advances
in the Theory of Automorphic Forms and their L-functions
(Cogdell volume) (2016), 339-350.
We show that if a 2-dimensional Artin representation corresponds to an
representation outside of a density 0 infinite set of places of a certain
form, then they correspond everywhere.
- Strong local-global
phenomena for Galois and automorphic representations
RIMS Kôkyûroku 1973, Modular forms and automorphic representations
An exposition of strong multiplicity one type results and refinements,
with an aim to explain the results of my Contemp. Math.
paper with Ramakrishnan.
Distinguishing graphs with zeta functions
and generalized spectra,
with Christina Durfee
Linear Algebra and its Applications 481 (2015), 54-82.
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
distinguishing graphs than the usual types of spectra studied.
How often should you clean your room?
with Krishnan (Ravi) Shankar
Discrete Mathematics & Theoretical Computer Science, Vol. 17, No. 1
We introduce and study a combinatorial optimization problem motivated by
the question, "How often should you clean your room?"
See also popular write-ups by Francis Woodhouse and
Local root numbers, Bessel models, and a conjecture of Guo and Jacquet,
with Masaaki Furusawa
Journal of Number Theory, Special Issue in Honor of
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.
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), 149-180.
As an application of our 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
In particular, one obtains the global Gross-Prasad Conjecture (a nonvanishing
theorem) for such representations.
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, Vol. 225, No. 1057 (2013), x+134pp.
We extend the fundamental lemma from our
American Journal paper below, as well as one due to Furusawa-Shalika,
to the full Hecke algebra.
- Nonunique factorization and principalization
in number fields
Proceedings of the AMS, Vol. 139, No. 9 (2011), 3025-3038.
This describes 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.
relative trace formula for a compact Riemann surface,
with Mark McKee and Eric Wambach [errata, corrected version]
International Journal of Number Theory, Vol. 7, No. 2 (2011), 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), 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 Mathematics Research Notices (IMRN) 2009, No. 1 (2009), 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.]
L-values and toric periods for GL(2)
RIMS Kôkyûroku 1617,
Automorphic Representations, Automorphic
Forms, L-functions and Related Topics (2008), 126-137.
This is basically an extended introduction to the above paper, ending with
an outline of the relative trace formula approach to proving special value
- Shalika periods on GL(2,D) and GL(4),
with Hervé Jacquet
Pacific Journal of Mathematics, Vol. 233, No. 2 (2007), 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), 10pp.
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
Ph.D. Thesis, Caltech, 2004, 81pp.
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
(by which I mean the mathematically- or scientifically- minded layman).
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
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.
PhD Students Supervised
Undergraduate Research Supervised
- Rank bias for elliptic curves, joint work with Thomas Pharis
(see my publication list)
Based on Thomas Pharis' 2019 Honors Thesis at the University of Oklahoma
asymptotic for the representation of integers as sums of triangular numbers,
by Atanas Atanasov, Rebecca Bellovin, Ivan Loughman-Pawelko, Laura Peskin and
Eric Potash. Involve, Vol. 1, No. 1 (2008) 111-121.
Carried out in the
2007 Summer Undergraduate Research Program at
Columbia University (See the program page for an open access version of
The correct solution
to Berelekamp's switching game, by Jordan Carlson and Daniel Stolarski. Disc. Math.
Vol. 287, No. 1-3 (2004), 145-150.
Carried out in the 2002 Caltech Freshman Summer Institute