Research and Publications
I am interested in number theory and automorphic forms. My doctoral research centered on Galois representations associated to automorphic forms.
Research:
- p-adic Families and Galois Representations for GSp(4) and GL(2) (pdf, ).
- Lagrangian hyperplanes in holomorphic symplectic varieties (arXiv, appendix, code). With Benjamin Bakker.
- Galois representations for holomorphic Siegel modular forms (pdf, ), to appear in Math. Annalen.
- Crystalline representations for $\operatorname{GL}(2)$ over quadratic imaginary fields (pdf, ), my thesis.
- Higher rank stable pairs on K3 surfaces (arXiv, ). With Benjamin Bakker.
- Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves (pdf ), Math. Comp. 78 (2009), no. 268, 2397--2425. With G. Grigorov, S. Patrikis, W. Stein, C. Tarniţǎ.
We prove local-global compatibility for Iwahori level Siegel modular forms by combining a previous
result (up to a quadratic twist) with p-adic families. We deduce information at p and l for two dimensional
Galois representations on quadratic imaginary fields.
We prove many cases of local-global compatibility (up to a quadratic twist) for
holomorphic Siegel modular forms.
If $\pi$ is an irreducible admissible regular algebraic cuspidal representation of $\textrm{GL}(2)$
over a quadratic imaginary field and $v$ is an unramified place of $K$ where $\pi_v$ and
$\pi_{v^c}$ are unramified principal series with distinct Satake parameters, we show that the
Galois representation associated to $\pi$ is crystalline at $v$.
Abstract. Virtual curve counts have been defined for threefolds
by integration against virtual classes on moduli spaces of stable
maps (Gromov-Witten theory), ideal sheaves (Donaldson-Thomas
theory), and stable pairs (Pandharipande-Thomas theory). The
first two theories are proven to be equivalent for toric threefolds,
and all three are conjecturally equivalent for arbitrary threefolds.
One may ask whether there is such a correspondence for surfaces.
In particular, the Gromov-Witten theory of $K3$ surfaces has recently been computed by Maulik, Pandharipande, and Thomas;
it is governed by quasimodular forms and is closely related to invariants obtained from the moduli spaces of rank $r = 0$ stable
pairs with $n = 1$ sections. We compute the Hodge polynomials of
the moduli spaces of stable pairs for higher rank $r \geq 0$ and level
$n \geq 1$, and explore the modularity properties and relationship to
Gromov-Witten theory.
Expository:
- The Birch and Swinnerton-Dyer conjecture for abelian varieties over number fields (pdf), my senior thesis at Harvard University, 2005.