Papers

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The trace of $T_2$ takes no repeated values
with Liubomir Chiriac
to appear in Indagationes Mathematicae
We prove that the trace of the Hecke operator $T_2$ acting on the space of weight $2k$ modular forms of level 1 takes no repeated values as $k$ varies. This answers a question of Vilardi and Xue, which arose in relation to Maeda's conjecture. Our proof combines 2-adic results from our previous paper (Newton Polygons of Hecke Operators) and a new application of classical bounds on linear forms on logarithms in the context of exponential sums with more than 2 terms.

Newton Polygons of Hecke Operators
with Liubomir Chiriac
Annales Mathématiques du Québec (2021)
In this computational paper we verify a truncated version of the Buzzard-Calegari conjecture on the Newton polygon of the Hecke operator $T_2$ for all large enough weights. We first develop a formula for computing $p$-adic valuations of exponential sums, which we then implement to compute 2-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard-Calegari polynomial has a vertex at $n\leq 15$, then it agrees with the Newton polygon of $T_2$ up to $n$.

$p$-adic $L$-functions of Hilbert cusp forms and the trivial zero conjecture
with Daniel Barrera and Mladen Dimitrov
Journal of the European Math. Soc. (2021)
In the first part of the paper, we show that any cohomological Hilbert modular cusp form which is non-critical and of nearly finite slope at $p$, in the sense that the local representation is not supercuspidal at any place above $p$, belongs to a unique $p$-adic family of maximal dimension. Further, using automorphic symbols for Hilbert modular varieties, we construct a functorial map sending a finite slope overconvergent cohomology class to a distribution over the Galois group of the maximal abelian $p$-ramified extension. These evaluation maps yield, in a unified fashion, both $p$-adic $L$-functions for nearly finite slope families and their improved counterparts.

In the second part of the paper we prove the exceptional zero conjecture at the central point for the Greenberg-Benois $p$-adic $\mathscr{L}$-invariant in the case of Hilbert modular forms which are in addition Iwahori spherical at places above p and have trivial central character. In the case of a multiple exceptional zero we go beyond the Greenberg-Stevens method and use the interplay between partial finite slope families and partially improved p-adic L-functions to establish the vanishing of many Taylor coefficients of the $p$-adic $L$-function of the family.

Comparing Hecke Coefficients of Automorphic Representations
with Liubomir Chiriac
Trans. Amer. Math. Soc. (2019)
We prove a number of statistical properties of Hecke coefficients for unitary cuspidal representations on $\operatorname{GL}(2)$ over number fields (unconditionally) and on $\operatorname{GL}(n)$ over number fields (conditionally, either assuming the Ramanujan conjecture, or the functoriality of $\pi\otimes\pi^\vee$). Using partial bounds on Hecke coefficients, properties of Rankin-Selberg $L$-functions, and instances of Langlands functoriality, we obtain bounds on the set of places where (linear combinations of) Hecke coefficients are bounded above (or below). We furthermore prove a number of consequences: we obtain an improved answer to a question of Serre about the occurrence of large Hecke eigenvalues of Maass forms ($|a_p|>1$ for density at least $0.00135$ set of primes), we prove the existence of negative Hecke coefficients over arbitrary number fields, and we obtain distributional results on the Hecke coefficients $a_v$ when $v$ varies in certain congruence or Galois classes. E.g., if $E$ is an elliptic curve without CM we show that $a_p(E)<0$ for a density $\geq \frac{1}{8}$ of primes $p\equiv a\pmod{n}$, or density $\geq \frac{1}{16}$ of primes of the form $p=m^2+27n^2$.

On symmetric power $\mathscr{L}$-invariants of Iwahori level Hilbert modular forms
with Rob Harron
American Journal of Mathematics 139 (2017), no. 6, 1605–1647
We compute the arithmetic $\mathscr{L}$-invariants (of Greenberg-Benois) of twists of symmetric powers of $p$-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on $p$-adic families of automorphic forms over symplectic and unitary groups. Combining these families with some explicit plethysm in the representation theory of $\operatorname{GL}(2)$, we construct global Galois cohomology classes with coefficients in the symmetric powers and provide formulae for the $\mathscr{L}$-invariants in terms of logarithmic derivatives of Hecke eigenvalues.

Lagrangian 4-planes in holomorphic symplectic varieties of K3([4])-type
with Benjamin Bakker
Central European Journal of Mathematics 12 (2014), no. 7, 952–975
We classify the cohomology classes of Lagrangian 4-planes $\mathbb{P}^4$ in a smooth manifold $X$ deformation equivalent to a Hilbert scheme of 4 points on a $K3$ surface, up to the monodromy action. Classically, the cone of effective curves on a $K3$ surface $S$ is generated by nonegative classes $C$, for which $(C,C)\geq0$, and nodal classes $C$, for which $(C,C)=-2$; Hassett and Tschinkel conjecture that the cone of effective curves on a holomorphic symplectic variety $X$ is similarly controlled by ``nodal'' classes $C$ such that $(C,C)=-\gamma$, for $(\cdot,\cdot)$ now the Beauville-Bogomolov form, where $\gamma$ classifies the geometry of the extremal contraction associated to $C$. In particular, they conjecture that for $X$ deformation equivalent to a Hilbert scheme of $n$ points on a $K3$ surface, the class $C=\ell$ of a line in a smooth Lagrangian $n$-plane $\mathbb{P}^n$ must satisfy $(\ell,\ell)=-\frac{n+3}{2}$. We prove the conjecture for $n=4$ by computing the ring of monodromy invariants on $X$, and showing there is a unique monodromy orbit of Lagrangian $4$-planes.

Galois representations for holomorphic Siegel modular forms

Mathematische Annalen 355 (2013), no. 1, 381–400
We prove local-global compatibility (up to a quadratic twist) of Galois representations associated to holomorphic Hilbert-Siegel modular forms in many cases (induced from Borel or Klingen parabolic), and as a corollary we obtain a conjecture of Skinner and Urban. For Siegel modular forms, when the local representation is an irreducible principal series we get local-global compatibility without a twist. We achieve this by proving a version of rigidity (strong multiplicity one) for $\operatorname{GSp}(4)$ using, on the one hand the doubling method to compute the standard $L$-function, and on the other hand the explicit classification of the irreducible local representations of $\operatorname{GSp}(4)$ over $p$-adic fields.

$p$-adic Families and Galois Representations for $\operatorname{GSp}(4)$ and $\operatorname{GL}(2)$

Mathematical Research Letters 19 (2012), no. 5, 1–10
We prove local-global compatibility for holomorphic Siegel modular forms with Iwahori level. In previous work we proved a weaker version of this result (up to a quadratic twist) and one of the goals of this article is to remove this quadratic twist by different methods, using $p$-adic families. We further study the local Galois representation at $p$ for nonregular holomorphic Siegel modular forms. Then we apply the results to the setting of modular forms on $\operatorname{GL}(2)$ over a quadratic imaginary field and prove results on the local Galois representation $\ell$, as well as crystallinity results at $p$.

Higher rank stable pairs on K3 surfaces
with Benjamin Bakker
Communications in Number Theory and Physics 6 (2012), no. 4, 805 – 847
We define and compute higher rank analogs of Pandharipande–Thomas stable pair invariants in primitive classes for $K3$ surfaces. Higher rank stable pair invariants for Calabi–Yau threefolds have been defined by Sheshmani using moduli of pairs of the form $\mathcal{O}^n\to \mathcal{F}$ for $\mathcal{F}$ purely one-dimensional and computed via wall-crossing techniques. These invariants may be thought of as virtually counting embedded curves decorated with a $(n−1)$-dimensional linear system. We treat invariants counting pairs $\mathcal{O}^n\to\mathcal{E}$ on a K3 surface for $\mathcal{E}$ an arbitrary stable sheaf of a fixed numerical type (“coherent systems”), whose first Chern class is primitive, and fully compute them geometrically. The ordinary stable pair theory of $K3$ surfaces is treated by Maulik-Pandharipande-Thomas; there they prove the KKV conjecture in primitive classes by showing the resulting partition functions are governed by quasimodular forms. We prove a “higher” KKV conjecture by showing that our higher rank partition functions are modular forms.

Crystalline representations for $\operatorname{GL}(2)$ over quadratic imaginary fields
Ph.D. thesis, Princeton, 2010
We show that Galois representations attached by Harris-Taylor-Soudry to regular algebraic cuspidal automorphic representations on $\operatorname{GL}(2)$ over a quadratic imaginary field (when the central character is a base-change from $\mathbb{Q}$) is crystalline at places above good primes $p$ as long as the Satake parameters at places above $p$ are distinct.

Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves
with Grigor Grigorov, Stefan Patrikis, William A. Stein, and Corina Tarniţǎ
Mathematics of Computation 78 (2009), no. 268, 2397–2425
We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over $ \mathbb{Q}$ of analytic ranks 0 and $ 1$. We apply our techniques to show that if $ E$ is a non-CM elliptic curve over $ \mathbb{Q}$ of conductor $ \leq 1000$ and rank 0 or $ 1$, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the $ L$-series is true for $ E$, up to odd primes that divide either Tamagawa numbers of $ E$ or the degree of some rational cyclic isogeny with domain $ E$. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or $ 1$, this completely verifies the full conjecture for these curves up to the primes excluded above.

Papers in preparation

Local-global compatibility at $\ell=p$ for Iwahori level Siegel-Hilbert modular forms

$\mathscr{L}$-invariants for $\operatorname{GSp}(2n)$ and $\operatorname{GL}(n)$ under algebraic representations

The Witten zeta function of projective varieties
with Benjamin Bakker

The Witten zeta functions of some spherical varieties
with Sam Evens

Expository

The Birch and Swinnerton-Dyer Conjecture for Abelian Varieties over Number Fields
My 2005 Harvard senior thesis written under the supervision of William Stein Errata
I explain Tate's proof of the fact that if the Birch and Swinnerton-Dyer conjecture is true for an abelian variety over a number field, it is also true for any isogenous abelian variety.