Research
[click to expand/collapse]
Topics: 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 objects (modular forms, automorphic representations, representations of
padic groups, Lfunctions, quaternion algebras,
algebraic groups, elliptic curves, genus 2 curves, ...).
Much of this is done with group theory,
representation theory, harmonic analysis, and/or computations.
I've also done a bit 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
download.

Counting modular forms by rationality field with Alex Cowan
submitted
We present data on weight 2 newforms of prime level < 2 million, and formulate conjectures about the distribution of weight 2 newforms with small rationality fields.

Local conductors bounds for modular abelian varieties
Acta Arithmetica, to appear.
We improve on the BrumerKramer bounds for local conductors of abelian
varieties of GL(2)type.

Moduli for rational genus 2 curves with real multiplication for discriminant 5 with Alex Cowan
Journal de Théorie des Nombres de Bordeaux, to appear.
The Hilbert modular surface Y(5) parametrizes prinicipally polarized
abelian surfaces with real multiplication (RM) by the ring of integers of
Q(√5).
We determine which moduli correspond to genus 2 curves defined over Q.

Mass formulas and Eisenstein congruences in higher rank
with Satoshi Wakatsuki
[preprint version]
Journal of Number Theory, Vol. 257 (2024), 249272.
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
forms.
The relevant Eisenstein series here are associated to minimal parabolic
subgroups.

Root number bias for newforms
Proceedings of the AMS, Vol. 151 (2023), 37213736.
We determine the number of newforms of a given weight and level with given
root number, and find that typically more newforms have root number +1 than
1. This generalizes
earlier results about levels which are squarefree or cubes of squarefree
numbers.

Rank bias for elliptic curves mod p
with Thomas Pharis
[preprint version]
[errata]
Involve, Vol. 15, No. 4 (2022), 709726.
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
modular forms. (See errata for a sign correction.)

Exact double averages of twisted Lvalues
Mathematische Zeitschrift, Vol. 302 (2022), 18211854.
We prove simple formulas for a double average of twisted central Lvalues
over both modular forms and twisting characters. This leads to generalizations
of stable (single) average formulas by MichelRamakrishnan and FeigonWhitehouse.
Errata: Springer introduced several errors during production:
the submission date was Jun 8 2020, not Jun 8 2022; the first and last names of authors are swapped in bib items 17, 20 and 29; a reference to the appendix correctly links to the appendix but the text says section 1

Refined Goldbach conjectures with primes in progressions
[preprint version]
Experimental Mathematics, Vol. 31, No. 1 (2022), 226232.
We present some refinements of Goldbach's conjectures by restricting
to primes in arithmetic progressions.

An onaverage Maedatype conjecture in the level aspect
[author version]
Proceedings of the AMS, Vol. 149, No. 4 (2021), 13731386.
[data]
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 Lfunctions.

Zeroes of quaternionic modular forms and central Lvalues
with Jordan Wiebe
[preprint version]
Journal of Number Theory, Vol. 217 (2020), 460494.
We study, theoretically and computationally, zeroes of
modular forms on definite quaternion algebras, and the relation to
non/vanishing of Lvalues.

The basis problem revisited
[preprint version]
Transactions of the AMS, Vol. 373, No. 7 (2020), 45234559.
We explicitly describe the JacquetLanglands 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
HijikataPizerShemanske for elliptic modular forms, and solves the
basis problem for Hilbert modular forms.

Rationality of Darmon points over genus fields of nonmaximal orders
with Matteo Longo and Yan Hu
[preprint version]
Annales mathématiques du Québec, Vol. 44, No. 1 (2020), 173195.
We extend work of BertoliniDarmonproving rationality of twists of
StarkHeegner points, aka Darmon points, for elliptic curvesfrom
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 Sideal classes
[preprint version]
Canadian Journal of Mathematics, Vol. 70, No. 5 (2018), 10761095.
We use quaternionic modular forms to prove various congruences mod 2
between modular forms with differing AtkinLehner eigenvalues. The proofs
are related to the distribution of AtkinLehner signs (making use of my
"Refined dimensions..." paper below) and
the notion of quaternionic Sideal classes.

Refined dimensions of cusp forms, and equidistribution and bias of signs
[preprint version]
[sage code]
Journal of Number Theory, Vol. 188 (2018), 117.
We give dimensions of new spaces of squarefree level with prescribed
AtkinLehner 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 Lvalues for GL(2n),
with Brooke Feigon and David Whitehouse
[preprint version]
Israel Journal of Mathematics, Vol. 225, No. 1 (2018), 223266.
Under some local hypotheses, we prove a relation between the nonvanishing
of twisted central Lvalues 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 JacquetLanglands
correspondence, Eisenstein congruences, and integral Lvalues in
weight 2 [errata, corrected version]
Mathematical Research Letters,
Vol. 24, No. 6 (2017), 17751795.
We use the JacquetLanglands correspondence to generalize
congruence results of Mazur to nonprime level and to
Hilbert modular forms.

Distinguishing
finite group characters and refined localglobal phenomena,
with Nahid Walji
[preprint version]
Acta Arithmetica, Vol. 179, No. 3 (2017), 277300.
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 Lfunctions
can agree in degree 2 or 3.

Test vectors and central Lvalues for GL(2),
with Daniel File and Ameya Pitale
[preprint version]
Algebra and Number Theory, Vol. 11, No. 2 (2017), 253318.
We extend work of Gross and Prasad on test vectors for GL(2) to cases of joint
ramification, and use this to generalize the Lvalue formula of my
2009 IMRN paper with Whitehouse, an averagevalue formula of
FeigonWhitehouse, and a nonvanishing mod p result of
MichelRamakrishnan.

A comparison of automorphic and
Artin Lseries of GL(2)type agreeing at degree one primes,
with Dinakar Ramakrishnan
Contemporary Mathematics 664, Advances
in the Theory of Automorphic Forms and their Lfunctions
(Cogdell volume) (2016), 339350.
We show that if a 2dimensional Artin representation corresponds to an
automorphic
representation outside of a density 0 infinite set of places of a certain
form, then they correspond everywhere.
 Strong localglobal
phenomena for Galois and automorphic representations
RIMS Kôkyûroku 1973, Modular forms and automorphic representations
(2015), 120130.
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
[arXiv version]
Linear Algebra and its Applications 481 (2015), 5482.
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.

How often should you clean your room?
with Krishnan (Ravi) Shankar
Discrete Mathematics & Theoretical Computer Science, Vol. 17, No. 1
(2015), 413442.
We introduce and study a combinatorial optimization problem motivated by
the question, "How often should you clean your room?"
See also popular writeups by Francis Woodhouse and
Jon Kujawa.

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), 150170.
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.

On
central critical values of the degree four Lfunctions for GSp(4):
a simple trace formula,
with Masaaki Furusawa
[preprint version]
Mathematische Zeitschrift, Vol. 277, No. 1 (2014), 149180.
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
hypotheses).
In particular, one obtains the global GrossPrasad Conjecture (a nonvanishing
theorem) for such representations.

On central critical values of the degree four Lfunctions for GSp(4): the fundamental lemma III,
with Masaaki Furusawa and Joseph Shalika
[preprint version]
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 FurusawaShalika,
to the full Hecke algebra.
 Nonunique factorization and principalization
in number fields
Proceedings of the AMS, Vol. 139, No. 9 (2011), 30253038.
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.
 A
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), 389429.
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
Lfunctions for GSp(4): the fundamental lemma II, with Masaaki
Furusawa [preprint version]
American Journal of Mathematics, Vol. 133, No. 1 (2011), 197233.
We propose a different kind of relative trace formula than FurusawaShalika
to relate central spinor Lvalues to Bessel periods, and prove the
corresponding fundamental lemma. This relative trace formula has several
advantages over the previous ones.
 Central Lvalues and toric periods
for GL(2), with David Whitehouse
International Mathematics Research Notices (IMRN) 2009, No. 1 (2009), 141191.
Using Jacquet's relative trace formula, we get a formula for the central value
of a GL(2) Lfunction, refining results of Waldspurger.
[Old version (Nov. 13, 2006). This uses a simpler trace formula but is much less general.]
 Central
Lvalues and toric periods for GL(2)
RIMS Kôkyûroku 1617,
Automorphic Representations, Automorphic
Forms, Lfunctions and Related Topics (2008), 126137.
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 Hervé Jacquet
[preprint version]
Pacific Journal of Mathematics, Vol. 233, No. 2 (2007), 341370.
Here we use a relative trace formula
to study period integrals, which yield results about exteriorsquare Lfunctions, 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).
 Fourdimensional Galois representations of solvable type and automorphic forms
[abstract]
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
the layman
(by which I mean the mathematically or scientifically minded layman).
 Modularity
of hypertetrahedral
representations
[preprint version]
Comptes Rendus Mathematique, Vol. 339, No. 2 (2004), 99102.
This proves a new case of modularity
for fourdimensional Galois representations induced from a nonnormal
quartic extension. In particular, one obtains examples of modular
representations which are not essentially selfdual.

A symplectic case of Artin's conjecture
Mathematical Research Letters, Vol. 10, No. 4 (2003), 483492.
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 mod p, by Kimball Martin and
Thomas Pharis
[preprint version]
[errata]
Involve, Vol. 15, No. 4 (2022), 709726.
Based on Thomas Pharis' 2019 Honors Thesis at the University of Oklahoma
 An
asymptotic for the representation of integers as sums of triangular numbers,
by Atanas Atanasov, Rebecca Bellovin, Ivan LoughmanPawelko, Laura Peskin and
Eric Potash. Involve, Vol. 1, No. 1 (2008) 111121.
Carried out in the
2007 Summer Undergraduate Research Program at
Columbia University (See the program page for an open access version of
their work.)

The correct solution
to Berelekamp's switching game, by Jordan Carlson and Daniel Stolarski. Disc. Math.
Vol. 287, No. 13 (2004), 145150.
Carried out in the 2002 Caltech Freshman Summer Institute
