The Number Theory and Algebraic Geometry (NTAG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry. The organizers for Fall 2023 are Katrina Honigs and Nathan Ilten. We normally meet in-person. For online editions, we use Zoom and distribute the link through the mailing list.

Time: |
August 6, 10:00-11:00 PDT |

Speaker: |
Mattia Talpo (Pisa) |

Title: |
Topological realization over $\mathbb{C}(\!(t)\!)$ via Kato-Nakayama spaces |

Abstract: |
I will talk about some joint work with Piotr Achinger, about a “Betti realization” functor for varieties over the formal punctured disk $\mathrm{Spec} \mathbb{C}(\!(t)\!)$, i.e. defined by polynomials with coefficients in the field of formal Laurent series in one variable over the complex numbers. We give two constructions producing the same result, one of them (the one that I'll talk about for most of the time) via “good models” over the power series ring $\mathbb{C}[\![t]\!]$ and the “Kato-Nakayama” construction in logarithmic geometry, that I will review during the talk. |

Time: |
July 30, 15:30-16:30PDT (this is a local edition) |

Speaker: |
Eugene Filatov (Simon Fraser University) |

Title: |
Brauer-Severi varieties associated to twists of the Burkhardt quartic |

Abstract: |
The Burkhardt quartic is a projective 3-fold which, geometrically, is birational to the moduli space of abelian surfaces with full level-3 structure. We study this moduli interpretation of the Burkhardt quartic in an arithmetic setting, over a general field k. As it turns out, some twists of the Burkhardt quartic have a nontrivial field-of-definition versus field-of moduli obstruction. Classically, if a twist has a k-rational point then the obstruction can be computed as the Brauer class of an associated conic.
Using representation theory, we show how to compute the obstruction without assuming the existence of a k-rational point, giving rise to an associated 3-dimensional Brauer-Severi variety rather than a conic. This Brauer-Severi variety itself has a related moduli interpretation.
This is joint work with Nils Bruin (my MSc supervisor). |

Time: |
July 23, 15:30-16:30PDT |

Speaker: |
Brendan Creutz (University of Canterbury) |

Title: |
Brauer-Manin obstructions on constant curves over global function fields |

Abstract: |
For a curve C over a global field K it has been conjectured that the Brauer-Manin obstruction explains all failures of the Hasse principle. I will discuss results toward this conjecture in the case of constant curves over a global function field, i.e. where C and D are curves over a finite field and we consider C over the function field of D. This is joint work with Felipe Voloch. |

Time: |
July 16, 15:30-16:30PDT |

Speaker: |
Bianca Viray (University of Washington) |

Title: |
Isolated points on modular curves |

Abstract: |
Faltings's theorem on rational points on subvarieties of abelian varieties can be used to show that all but finitely many algebraic points on a curve arise in families parametrized by $\mathbb{P}^1$ or positive rank abelian varieties; we call these finitely many exceptions isolated points. We study how isolated points behave under morphisms and then specialize to the case of modular curves. We show that isolated points on $X_1(n)$ push down to isolated points on a modular curve whose level is bounded by a constant that depends only on the j-invariant of the isolated point. This is joint work with A. Bourdon, O. Ejder, Y. Liu, and F. Odumodu. |

Time: |
July 9, 15:30-16:30PDT |

Speaker: |
Anthony Várilly-Alvarado (Rice University) |

Title: |
Rational surfaces and locally recoverable codes |

Abstract: |
Motivated by large-scale storage problems around data loss, a budding branch of coding theory has surfaced in the last decade or so, centered around locally recoverable codes. These codes have the property that individual symbols in a codeword are functions of other symbols in the same word. If a symbol is lost (as opposed to corrupted), it can be recomputed, and hence a code word can be repaired. Algebraic geometry has a role to play in the design of codes with locality properties. In this talk I will explain how to use algebraic surfaces birational to the projective plane to both reinterpret constructions of optimal codes already found in the literature, and to find new locally recoverable codes, many of which are optimal (in a suitable sense). This is joint work with Cecília Salgado and Felipe Voloch. |

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. |

Time: |
June 25, 15:30-16:30PDT |

Speaker: |
Avinash Kulkarni (Dartmouth) |

Title: |
pNumerical Linear Algebra |

Abstract: |
In this talk, I will present new algorithms, based on ideas from numerical analysis, for efficiently computing the generalized eigenspaces of a square matrix with finite precision p-adic entries. I will then discuss how these eigenvector methods can be used to compute the (approximate) solutions to a zero-dimensional polynomial system.
(Some content ongoing work with T. Vaccon) |

Time: |
June 11, 15:30-16:30PDT |

Speaker: |
Jake Levinson (University of Washington) |

Title: |
Boij-Söderberg Theory for Grassmannians |

Abstract: |
The Betti table of a graded module over a polynomial ring encodes much of its structure and that of the corresponding sheaf on projective space. In general, it is hard to tell which integer matrices can arise as Betti tables. An easier problem is to describe such tables up to positive scalar multiple: this is the "cone of Betti tables". The Boij-Söderberg conjectures, proven by Eisenbud-Schreyer, gave a beautiful description of this cone and, as a bonus, a "dual" description of the cone of cohomology tables of sheaves.
I will describe some extensions of this theory, joint with Nicolas Ford and Steven Sam, to the setting of GL-equivariant modules over coordinate rings of matrices. Here, the dual theory (in geometry) concerns sheaf cohomology on Grassmannians. One theorem of interest is an equivariant analog of the Boij-Söderberg pairing between Betti tables and cohomology tables. This is a bilinear pairing of cones, with output in the cone coming from the "base case" of square matrices, which we also fully characterize. |

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: |
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. |

For previous semesters, see archived schedules.