Contributed Talks Schedule
Week 3
Monday 7/28 - NATRS 140:
Chair: Ignacio Rojas
1:30 Zeyu Liu
2:05 Andrés Jaramillo Puentes
2:50 Jun Yong Park
3:25 Yanshuai Qin
4:10 Adam Logan
4:45 Andrew Kobin
Tuesday 7/29 - NATRS 113:
Chair: Chris Peterson
1:30 Elyes Boughattas
2:05 Emma Brakkee
2:50 Haitao Zou
3:25 Simone Coccia
4:10 Runxuan Gao
4:45 Damián Gvirtz-Chen
Wednesday 7/30 - YATES 104:
Chair: Mark Shoemaker
1:30 Stephen McKean
2:05 Remy van Dobben de Bruyn
2:50 Daniel Li-Huerta
3:25 Gordon Heier
4:10 Gyujin Oh
Thursday 7/31 - PATH 101:
Chair: Eamon Gannon
1:30 Kwangho Choiy
2:05 Sebastian Casalaina-Martin
2:50 Marc Abboud
3:25 Hyun Jong Kim
Titles and Abstracts
Zeyu Liu. A stacky approach to prismatic crystals
In this talk, we will explain how prismatic crystals can be realized as certain D-modules via the stacky approach of Drinfeld and Bhatt–Lurie, thereby yielding a classification of the former.
Andrés Jaramillo Puentes. Computations of Motivic Gromov-Witten Invariants
Recently, Kass–Levine–Solomon–Wickelgren proved the invariance of the motivic count of curves on del Pezzo surfaces, defining a Gromov-Witten-type invariant valued in the Grothendieck–Witt group of an arbitrary base field. In this talk, I will present a correspondence theorem that allows the computation of such invariants for curves in toric del Pezzo surfaces passing through k-rational points and points defined over arbitrary quadratic field extensions. This is joint work with Hannah Markwig, Sabrina Pauli, and Felix Röhrle.
Jun Yong Park. Totality & Inner arithmetic of rational points on moduli stacks
It is natural to want to count elliptic curves over a global field such as the field Q of rational numbers or the field Fq(t) of rational functions over the finite field Fq. To this end, we consider the fact that each E/K corresponds to a K-rational point on the fine moduli stack Mbar_{1,1} of stable elliptic curves, which in turn corresponds to a rational curve on Mbar_{1,1}. In this talk, I will explain the exact counting formula for all elliptic curves over k(t) via motivic methods along with a geometric explanation for the origin of lower order main terms based on the Height moduli framework (joint work with Dori Bejleri and Matthew Satriano).
Yanshuai Qin. On p-torsions of geometric Brauer groups
For a smooth projective variety $X$ over a finitely generated field in positive characteristic $p > 0$, we show that the Tate conjecture for divisors on $X$ is equivalent to the boundedness of the $p$-torsions in the Galois-fixed part of the geometric Brauer group of $X$, generalizing well-known results concerning the prime-to-$p$ part of the geometric Brauer group. This is joint work with Zhenghui Li.
Adam Logan. Kodaira dimension of Hilbert modular threefolds
Following a method introduced by Thomas-Vasquez and developed by Grundman, we prove that many Hilbert modular threefolds of geometric genus $0$ and $1$ are of general type, and that some are of nonnegative Kodaira dimension. The new ingredient is a detailed study of the geometry and combinatorics of totally positive integral elements $x$ of a fractional ideal $I$ in a totally real number field $K$ with the property that $\text{tr}\, xy < \min I\,\text{tr}\, y$ for some $y \gg 0 \in K$.
Andrew Kobin. Stacky curves and rings of mod p modular forms
We extend work of Voight and Zureick-Brown to compute the log canonical ring of a stacky curve over a field of characteristic $p > 0$. This in turn allows us to compute rings of mod p modular forms. Our approach also reveals that for $p = 2,3$, there are infinitely many levels $N$ for which there are weight $2$ modular forms of level $\Gamma_{0}(N)$ that do not lift to characteristic $0$. This is joint work with David Zureick-Brown.
Elyes Boughattas. Unirationality and R-equivalence on conic bundle surfaces
Over a finite field k, Yanchevskiĭ asked whether a surface X is unirational when f:X->P^1_k is a conic bundle. In 1996, Mestre had supplied a positive answer when the cardinal of k is much larger than the degree of the "bad locus" of f. I will present a recent result where I answer Yanchevskiĭ's question when the "bad fibres" of f lie above rational points of P^1_k. As a bonus, and under the same conditions, the method we use proves that X has a unique R-equivalence class. These results hold more generally over quasi-finite fields. (arXiv:2410.19686v2)
Emma Brakkee. Bounding Brauer groups of K3 surfaces using moduli spaces
For a K3 surface over a number field, the transcendental part of its Brauer group is finite. It was shown by Cadoret-Charles that the size of its l-primary torsion is uniformly bounded for K3 surfaces in one-dimensional families. We give a new proof of this result for one-dimensional families of K3 surfaces with a polarization by a fixed lattice. To be precise, we construct moduli spaces of K3 surfaces with a lattice polarization and a Brauer class, and use the geometry of their complex points to prove boundedness of Brauer groups for the K3 surfaces they parametrize. I will explain the construction and sketch the proof of our boundedness result. This is joint work in with D. Bragg and A. Várilly-Alvarado.
Haitao Zou. A p-adic analogue of Nagai’s conjecture for hyper-Kähler varieties
Hyper-Kähler (HK) varieties, as higher-dimensional analogues of K3 surfaces, are known to have much of their geometry governed by their weight-two Hodge structures. Inspired by this perspective, Yasunari Nagai proposed a conjecture asserting that the unipotency of monodromy operators on complex HK varieties is entirely determined by their action on the second cohomology. In this talk, I will introduce a p-adic analogue of Nagai’s conjecture for HK varieties over local fields, and confirm this for all four known deformation types. I will also present some arithmetic applications of this conjecture for HK varieties.This is joint work with Kazuhiro Ito, Tetsushi Ito, Teruhisa Koshikawa, and Teppei Takamatsu.
Simone Coccia. Density of integral points in the Betti moduli of quasi-projective varieties
Let Y be a smooth quasi-projective complex variety equipped with a simple normal crossings compactification. We show that integral points are potentially dense in the (relative) character varieties parametrizing SL_2-local systems on Y with fixed algebraic integer traces along the boundary components. The proof proceeds by using work of Corlette-Simpson to reduce to the case of Riemann surfaces, where we produce an integral point with Zariski-dense orbit under the mapping class group. This is joint work with Daniel Litt.
Runxuan Gao. Zariski dense exceptional sets in Manin's conjecture: dimension 2
On rationally connected varieties over number fields, an exceptional set is a subset where the number of rational points of bounded height grows faster than expected. While varieties of dimension ≥3 with a Zariski dense exceptional set have been known for some time, we provide the first such examples in dimension 2 and study them systematically.
Damián Gvirtz-Chen. Non-unirational Hilbert Irreducibility and the Inverse Galois Problem
We establish non-unirational generalisations of Hilbert Irreducibility for low-discriminant Hilbert Modular Surfaces. As an application, we realise new cases of the Inverse Galois Problem for certain groups of type PSL_2(F_p^2). Joint work with J. Demeio.
Stephen McKean. Quadratic Segre indices
Over C, there is a finite number of lines on a general degree 2n-1 hypersurface in projective n+1 space. Over R, the total number of real lines can vary, but Finashin--Kharlamov and Okonek--Teleman proved that the signed count of real lines is always (2n-1)!!. In this talk, I will discuss joint work with Felipe Espreafico and Sabrina Pauli, in which we extend this signed count of lines to arbitrary fields of characteristic not 2.
Remy van Dobben de Bruyn. A categorical independence of ℓ conjecture
In this talk, I formulate a new categorical independence of ℓ conjecture for constructible sheaves on varieties over finite fields, linking Deligne's companions conjecture (proven by Drinfeld on smooth varieties) with the independence of ℓ conjecture on Betti numbers (still widely open). I show that this new conjecture is weaker than existing conjectures, is true at the level of K₀, and can be verified in special cases, such as toric varieties and abelian varieties.
Daniel Li-Huerta. A global Hartl–Pink curve
The "fundamental curve" of arithmetic over nonarchimedean local fields is the Fargues–Fontaine curve. You'd think the "fundamental curve" of arithmetic over global function fields is... just the associated curve over Fq. In this talk, I will instead advocate for a different "curve", in the spirit of the Fargues–Fontaine curve. I will explain how its moduli stack of bundles is the function field analog of Igusa stacks, as well as the perspective it gives on Langlands duality.
Gordon Heier. Generalizations of Schmidt's Subspace Theorem and applications to diophantine (quasi-)hyperbolicity problems
The Theorem of Roth is a fundamental result concerning the diophantine approximation to algebraic numbers. There is a higher-dimensional version due to Schmidt, called the Subspace Theorem, which concerns diophantine approximation to hyperplanes in general position in projective space. The geometric nature of the Subspace Theorem has given rise to numerous generalizations over the years by many authors. In this talk, we will present generalized versions of Schmidt’s Subspace Theorem concerning diophantine approximation to closed subschemes. A key feature of our results i the use of Seshadri constants from algebraic geometry. Our results are particularly elegant when the closed subschemes are in general position, but they also apply to the case of subgeneral position, providing a joint generalization of Schmidt's theorem with seminal inequalities of Nochka. Time permitting, we will also address applications to diophantine (quasi-)hyperbolicity problems, i.e., to the degeneracy of integral points on varieties in the complement of suitably positive divisors. This is joint work with Aaron Levin.
Gyujin Oh. Weight 1 modular forms and obstruction to geometricity
The arithmetic of weight 1 modular forms ("irregular" case) is much more delicate than that of weight >2 ("regular" case). We explain how this is related to the non-geometric extensions of geometric objects in the p-adic context and the archimedean context.
Kwangho Choiy. On reducibility of parabolic induction over weakly unramified characters
Given a p-adic group its relative Weyl group with respect to a split torus acts on weakly unramified characters and this forms a complex affine variety. Viewing supercuspidal supports of representations having non-trivial parahoric-fixed vectors as an element in the complex affine variety, we shall present some invariant properties of parabolic inductions within a finite subset of the variety and discuss their behavior between inner forms. This work also pursues an invariant within non-discrete tempered L-packets.
Sebastian Casalaina-Martin. Regular homomorphisms, with a twist
I will discuss some recent results, joint with Jeff Achter and Charles Vial, regarding regular homomorphisms, and families of cycles over bases that do not admit a point over the field of definition. This can be used to provide an obstruction to the existence of algebraic cycles defined over the base field. I will then connect this obstruction to some recent results of Hassett–Tschinkel and Benoist–Wittenberg on rationality of threefolds.
Marc Abboud. On the rigidity of periodic points for loxodromic automorphisms of affine surfaces
We show the following result. Let f,g be two automorphisms of a normal affine surface with dynamical degree >1. If f and g share a Zariski dense set of periodic points, then they have the same periodic points. This result fits into the category of unlikely intersections problem. The main idea is to construct canonical heights for such automorphisms and use arithmetic equidistribution theorems. I will also discuss some special affine surfaces where we prove the stronger result that f and g must share a common iterate.
Hyun Jong Kim. Independence-of-ell for attributes of Tannakian monodromy groups of perverse sheaves on commutative algebraic groups
Abstract: In ongoing work with Chris Hall, we start with a connected, commutative algebraic group G over a finite field Fq along with nice perverse sheaves M1 and M2 on G with respectively ell1-adic and ell2-adic coefficients with essentially the same Frobenius eigenvalues at stalks. M1 and M2 each generate (arithmetic and geometric) neutral tannakian categories, associated to which are tannakian groups G1 and G2, which are affine algebraic groups over Qell1bar and Qell2bar. Our goal is to prove that attributes, such as the group of components, of G1 and G2 are equal to each other. I will talk about progress that we have made towards such results in some cases.
Locations
Contributed talk sessions will happen in the following classrooms. Here is a link to the CSU campus map, and you may click on each classroom to see its location.