Current contacts: Jaclyn Lang, Catherine Hsu, Ian Whitehead, and Djordje Milicevic
The Philadelphia Area Number Theory Seminar rotates between Bryn Mawr, Swarthmore, and Temple. In Fall 2022, we meet on Tuesday afternoons, usually with tea at 3pm and then the talk 3:30-5pm. Please pay careful attention to the times and locations of the talks as they change from week to week! If you would like to be added to our mailing list or if you are interested in being a speaker, please contact one of the organizers. In future semesters, we anticipate that the seminar will be on Wednesday afternoons.
Click on title for abstract.
Souparna Purohit, University of Pennsylvania
Given an arithmetic variety $\mathscr{X}$ and a hermitian line bundle $\overline{\mathscr{L}}$, the arithmetic Hilbert-Samuel theorem describes the asymptotic behavior of the co-volumes of the lattices $H^0(\mathscr{X}, \mathscr{L}^{\otimes k})$ in the normed spaces $H^0(\mathscr{X}, \mathscr{L}^{\otimes k}) \otimes \mathbb{R}$ as $k \to \infty$. Using his work on quasi-filtered graded algebras, Chen proved a variant of the arithmetic Hilbert-Samuel theorem which studies the asymptotic behavior of the successive minima of the lattices above. Chen's theorem, however, requires that the metric on $\overline{\mathscr{L}}$ is continuous, and hence does not apply to automorphic vector bundles for which the natural metrics are often singular. In this talk, we discuss a version of Chen's theorem for the line bundle of modular forms for a finite index subgroup $\Gamma \subseteq \text{PSL}_2(\mathbb{Z})$ endowed with the logarithmically singular Petersson metric. This generalizes work of Chinburg, Guignard, and Soul\'{e} addressing the case $\Gamma = \text{PSL}_2(\mathbb{Z})$.
Steven Groen, Lehigh University
The Schottky problem is a classical problem that asks which Abelian varieties are isomorphic to the Jacobian of a (smooth) curve. If the dimension exceeds 3, not every Abelian variety can be a Jacobian. In characteristic p, there are additional tools that shed light on this question. In particular, the Ekedahl-Oort stratification partitions Abelian varieties by their p-torsion group scheme. An example of this is the distinction between ordinary elliptic curves and supersingular elliptic curves. The Ekedahl-Oort stratification leads to the following question: which p-torsion group schemes arise from Jacobians of (smooth) curves? Although this question is still wide open, I will present some progress on it, in particular when the curves in question are Artin-Schreier covers. Part of this is joint work with Huy Dang.
Akshay Venkatesh, Institute of Advanced Study
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Preston Wake, Michigan State University
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