Quarantined Number Theory and Algebraic Geometry Seminar

The Number Theory and Algebraic Geometry (NTAG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry. It is co-organized by Nathan Ilten and Nils Bruin. For Summer 2020, we will be meeting virtually via zoom. The link for the meeting is distributed via our local mailing list.

For previous semesters, see archived schedules.

Schedule for Summer 2020

We normally meet on Thursdays 15:30-16:30PDT, but we are happy to reschedule to better suit the timezone of the speaker. Adjusted times are displayed in red.

Upcoming talks

Time: June 4, 15:30-16:30PDT
Speaker: Türkü Özlüm Çelik (Leipzig University)
Title: The Dubrovin threefold of an algebraic curve
Abstract: The solutions to the Kadomtsev-Petviashvili equation that arise from a fixed complex algebraic curve are parametrized by a threefold in a weighted projective space, which we name after Boris Dubrovin. Current methods from nonlinear algebra are applied to study parametrizations and defining ideals of Dubrovin threefolds. We highlight the dichotomy between transcendental representations and exact algebraic computations.

This is joint work with Daniele Agostini and Bernd Sturmfels.

Time: July 2, 15:30-16:30PDT
Speaker: Daniele Turchetti (Dalhousie)
Title: Moduli spaces of Mumford curves over $\mathbb Z$
Abstract: Schottky uniformization is the description of an analytic curve as the quotient of an open dense subset of the projective line by the action of a Schottky group. All complex curves admit this uniformization, as well as some $p$-adic curves, called Mumford curves. In this talk, I present a construction of universal Mumford curves, analytic spaces that parametrize both archimedean and non-archimedean uniformizable curves of a fixed genus. This result relies on the existence of suitable moduli spaces for marked Schottky groups, that can be built using the theory of Berkovich spaces over rings of integers of number fields due to Poineau.

After introducing Poineau's theory from scratch, I will describe universal Mumford curves and explain how these can be used as a framework to study the Tate curve and to give higher genus generalizations of it. This is based on joint work with Jérôme Poineau.

Schedule for available slots

While the regularly scheduled slots below are filled, we can schedule extra ones if appropriate.

Date Speaker
May 7, 15:30PDT Katrina Honigs (University of Oregon)
May 14, 10am PDT Christophe Ritzenthaler (Rennes)
May 21, 10am PDT Fabien Pazuki (Copenhagen)
May 28, 15:30PDT Nathan Ilten (SFU)
June 4, 15:30PDT Türkü Özlüm Çelik (Leipzig University)
June 11, 15:30PDT Jake Levinson (University of Washington)
June 18, 15:30PDT Pedro Mendoza (SFU)
June 25, 15:30PDT Avinash Kulkarni (Dartmouth)
July 2, 15:30PDT Daniele Turchetti (Dalhousie)
July 9, 15:30PDT Anthony Várilly-Alvarado (Rice University)
July 16, 15:30PDT Bianca Viray (University of Washington)
July 23, 15:30PDT Brendan Creutz (University of Canterbury)
July 30, 15:30PDT Eugene Filatov (SFU)
August 6, 10am Mattia Talpo (Pisa)

Past talks

Time: May 28, 15:30-16:30PDT
Speaker: Nathan Ilten (SFU)
Title: Fano schemes for complete intersections in toric varieties
Abstract: The study of the set of lines contained in a fixed hypersurface is classical: Cayley and Salmon showed in 1849 that a smooth cubic surface contains 27 lines, and Schubert showed in 1879 that a generic quintic threefold contains 2875 lines. More generally, the set of k-dimensional linear spaces contained in a fixed projective variety X itself is called the k-th Fano scheme of X. These Fano schemes have been studied extensively when X is a general hypersurface or complete intersection in projective space.

In this talk, I will report on work with Tyler Kelly in which we study Fano schemes for hypersurfaces and complete intersections in projective toric varieties. In particular, I'll give criteria for the Fano schemes of generic complete intersections in a projective toric variety to be non-empty and of "expected dimension". Combined with some intersection theory, this can be used for enumerative problems, for example, to show that a general degree (3,3)-hypersurface in the Segre embedding of $\mathbb{P}^2\times \mathbb{P}^2$ contains exactly 378 lines.

Time: May 21, 10:00-11:00PDT
Speaker: Fabien Pazuki (Copenhagen)
Title: Regulators of number fields and abelian varieties
Abstract: In the general study of regulators, we present three inequalities. We first bound from below the regulators of number fields, following previous works of Silverman and Friedman. We then bound from below the regulators of Mordell-Weil groups of abelian varieties defined over a number field, assuming a conjecture of Lang and Silverman. Finally we explain how to prove an unconditional statement for elliptic curves of rank at least 4. This third inequality is joint work with Pascal Autissier and Marc Hindry. We give some corollaries about the Northcott property and about a counting problem for rational points on elliptic curves.

Time: May 14, 10:00-11:00PDT
Speaker: Christophe Ritzenthaler (Rennes)
Title: Jacobians in the isogeny class of $E^g$
Abstract: Let $E$ be an ordinary elliptic curve over a finite field $\mathbb{F}_q$ such that $R=\mathrm{End}(E)$ is generated by the Frobenius endomorphism. There is an equivalence of categories which associates to each abelian variety $A$ in the isogeny class of $E^g$ an $R$-lattice $L$ of rank $g$. Given $L$ (with a hermitian form describing a polarization $a$ on $A$), we show how to make $(A,a)$ concrete, i.e. we give an embedding of $(A,a)$ into a projective space by computing its algebraic theta constants. Using these data and an algorithm to compute Siegel modular forms algebraically, we can decide when $(A,a)$ is a Jacobian over $\mathbb{F}_q$ when $g \leq 3$ (and over $\bar{\mathbb{F}}_q$ when $g=4$). We illustrate our algorithms with the problem of constructing curves over $\mathbb{F}_q$ with many rational points.

Joint work with Markus Kirschmer, Fabien Narbonne and Damien Robert

Time: May 7, 15:30-16:30PDT
Speaker: Katrina Honigs (University of Oregon)
Title: An obstruction to weak approximation on a Calabi-Yau threefold
Abstract: The study of Q-rational points on algebraic varieties is fundamental to arithmetic geometry. One of the few methods available to show that a variety does not have any Q-points is to give a Brauer-Manin obstruction. Hosono and Takagi have constructed a class of Calabi-Yau threefolds that occur as a linear section of a double quintic symmetroid and gave a detailed analysis of them as complex varieties in the context of mirror symmetry. These threefolds come tantalizingly equipped with a natural Brauer class, and their construction can be used to produce varieties over Q as well. In forthcoming work with Hashimoto, Lamarche and Vogt, we analyze these threefolds and their Brauer class over Q and give a condition under which the Brauer class obstructs weak approximation, though it cannot obstruct the existence of Q-rational points.