1. Distribution of local signs of modular forms and murmurations of Fourier coefficients
   submitted
   We determine how local root numbers of modular forms are distributed, and show the local root numbers are correlated with Fourier coefficients. This is an analogue of earlier work on global root numbers. We also conjecture the existence of murmurations beyond the context of global root numbers.

2. Generic models for genus 2 curves with real multiplication with Alex Cowan and Sam Frengley
   submitted
   We give generic models for genus 2 curves with real multiplication by the ring of integers in Q(√D) for all fundamental discriminants D < 40, and D = 44, 53, 61 by algorithmically minimizing conics over function fields. This is a sequel to my Bordeaux paper with Cowan.

3. Counting modular forms by rationality field with Alex Cowan
   submitted
   We make conjectures about the distribution of degrees and rationality fields of weight 2 newforms, and provide some evidence toward these conjectures.

4. Moduli for rational genus 2 curves with real multiplication for discriminant 5 with Alex Cowan
   Journal de Théorie des Nombres de Bordeaux, Vol. 32, No. 2 (2024), 493-525.
   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.

5. Local conductors bounds for modular abelian varieties [preprint version]
   Acta Arithmetica, Vol. 212, No. 4 (2024), 325-336.
   We improve on the Brumer-Kramer bounds for local conductors of abelian varieties of GL(2)-type.

6. Mass formulas and Eisenstein congruences in higher rank with Satoshi Wakatsuki [preprint version]
   Journal of Number Theory, Vol. 257 (2024), 249-272.
   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 extend some earlier results on Hilbert modular forms. The relevant Eisenstein series here are associated to minimal parabolic subgroups.

7. Root number bias for newforms
   Proceedings of the AMS, Vol. 151 (2023), 3721-3736.
   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.

8. Rank bias for elliptic curves mod p with Thomas Pharis [preprint version] [errata]
   Involve, Vol. 15, No. 4 (2022), 709-726.
   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.)

9. Exact double averages of twisted L-values
   Mathematische Zeitschrift, Vol. 302 (2022), 1821-1854.
   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.
   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

10. Refined Goldbach conjectures with primes in progressions [preprint version]
    Experimental Mathematics, Vol. 31, No. 1 (2022), 226-232.
    We present some refinements of Goldbach's conjectures by restricting to primes in arithmetic progressions.

11. An on-average Maeda-type conjecture in the level aspect [author version]
    Proceedings of the AMS, Vol. 149, No. 4 (2021), 1373-1386. [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 L-functions.

12. Zeroes of quaternionic modular forms and central L-values with Jordan Wiebe [preprint version]
    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.

13. The basis problem revisited [preprint version]
    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.

14. Rationality of Darmon points over genus fields of non-maximal orders with Matteo Longo and Yan Hu [preprint version]
    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.

15. Congruences for modular forms mod 2 and quaternionic S-ideal classes [preprint version]
    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.

16. Refined dimensions of cusp forms, and equidistribution and bias of signs [preprint version] [sage code]
    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.

17. Periods and nonvanishing of central L-values for GL(2n), with Brooke Feigon and David Whitehouse [preprint version]
    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.

18. 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.

19. Distinguishing finite group characters and refined local-global phenomena, with Nahid Walji [preprint version]
    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.

20. Test vectors and central L-values for GL(2), with Daniel File and Ameya Pitale [preprint version]
    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 Michel-Ramakrishnan.

21. 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 automorphic representation outside of a density 0 infinite set of places of a certain form, then they correspond everywhere.

22. Strong local-global phenomena for Galois and automorphic representations
    RIMS Kôkyûroku 1973, Modular forms and automorphic representations (2015), 120-130.
    An exposition of strong multiplicity one type results and refinements, with an aim to explain the results of my Contemp. Math. paper with Ramakrishnan.

23. Distinguishing graphs with zeta functions and generalized spectra, with Christina Durfee [arXiv version]
    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 effective at distinguishing graphs than the usual types of spectra studied.

24. How often should you clean your room? with Krishnan (Ravi) Shankar
    Discrete Mathematics & Theoretical Computer Science, Vol. 17, No. 1 (2015), 413-442.
    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 Jon Kujawa.

25. 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), 150-170.
    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.

26. On central critical values of the degree four L-functions for GSp(4): a simple trace formula, with Masaaki Furusawa [preprint version]
    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 hypotheses). In particular, one obtains the global Gross-Prasad Conjecture (a nonvanishing theorem) for such representations.

27. 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, 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.

28. 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.

29. 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), 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.

30. 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.

31. 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.]

32. Central 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 formulas.

33. Shalika periods on GL(2,D) and GL(4), with Hervé Jacquet [preprint version]
    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).

34. 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).

35. Four-dimensional 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).

36. Modularity of hypertetrahedral representations [preprint version]
    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.

37. 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

• Jordan Wiebe (PhD 2019, subsequent position: Lockheed Martin)
  Thesis: Arithmetic in Quaternion Algebras and Quaternionic Modular Forms
• Siddharth Ramakrishnan (current)
• Sayantan Santra (current)

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), 709-726.
  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 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 their work.)

• 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

Notes on Number Theory and Representation Theory

• An introduction to (principal) L-functions and L-groups, from a prep session for grad students at TORA VIII (19pp, Mar 2017).

• Sums of squares, sums of cubes, and modern number theory, a sort of survey, aimed at graduate students (27pp, 2015; revised Jun 2016).

• A brief overview of modular and automorphic forms, aimed at graduate students (12pp, 2010; revised Jun 2016).

• My thesis for the layman, an attempt to vaguely explain what I was working on to my friends/undergrad students from Caltech (4pp, 2004).

• Langlands, Tunnell, Wiles and Fermat. This is an attempt to very briefly (and informally) explain how L-functions and automorphic forms/representations are involved in the proof of Fermat's Last Theorem (4pp, 2004).

• Langlands' Conjecture for the Tetrahedral and Octahedral Cases, a short introduction to Langlands' reciprocity conjecture with an exposition of the proof in the tetrahedral and octahedral cases, i.e., the Langlands-Tunnell Theorem (7pp, 2002).

• Representations of S_3, A_4 and S_4, a simple exercise to write the irreducible representations as induced from one-dimensionals (2pp).

Notes on Graph Theory and Algebraic Combinatorics

• Designs and Codes: Planes, Difference Sets and Hadamard Matrices, from a talk to general math undergrads on some research areas in algebraic combinatorics (5pp, revised 2009).

• A Brief Introduction to Coding Theory, aimed at introducing my Caltech Freshman Summer Institute kids to a research project on Berlekamp's light bulb game (4pp, 2002). You can see their work here.

• Difference Sets with Group Characters, a quick proof, shown to me by John Dillon, of Maschietti's theorem, which gave us a new construction for an infinite class of difference sets (6pp, 1997).