# A variety that cannot be dominated by one that lifts.

## Speaker:

Remy van Dobben de Bruyn

## Institution:

IAS/Princeton University

## Time:

Thursday, April 22, 2021 - 3:00pm

## Location:

Zoom https://uci.zoom.us/j/99706368574

The recent proofs of the Tate conjecture for K3 surfaces over finite fields start by lifting the surface to characteristic 0. Serre showed in the sixties that not every variety can be lifted, but the question whether every motive lifts to characteristic 0 is open. We give a negative answer to a geometric version of this question, by constructing a smooth projective variety that cannot be dominated by a smooth projective variety that lifts to characteristic 0.

# On applications of arithmetic geometry in commutative algebra and algebraic geometry.

Jakub Witaszek

## Institution:

University of Michigan, Ann Arbor

## Time:

Thursday, May 6, 2021 - 3:00pm

## Location:

Zoom https://uci.zoom.us/j/96117950184

In this talk, I will discuss how recent developments in arithmetic geometry (for example pertaining to perfectoid spaces) led to significant new discoveries in commutative algebra and algebraic geometry in mixed characteristic.

# A deformational semistable p-adic Hodge conjecture

Oli Gregory

## Institution:

University of Exeter

## Time:

Thursday, June 3, 2021 - 3:00pm

## Location:

Zoom meeting : https://uci.zoom.us/j/94130179823

Let k be a perfect field of characteristic p>0 and let X be a proper scheme over W(k) with semistable reduction. I shall define a log-Chow group for the special fibre X_k and give an interpretation in terms of logarithmic Milnor K-theory. Then, by gluing a logarithmic variant of the Suslin-Voevodsky motivic complex to a log-syntomic complex along the logarithmic Hyodo-Kato Hodge-Witt sheaf, I will prove that an element of the r-th log-Chow group of X_k formally lifts to the continuous log-Chow group of X if and only if it is “Hodge” (i.e. its log-crystalline Chern class lands in the r-th step of the Hodge filtration of the generic fibre of X under the Hyodo-Kato isomorphism). This simultaneously generalises a result of Yamashita (which is the case r=1), and of Bloch-Esnault-Kerz (which is the case of good reduction). This is joint work with Andreas Langer.

# Arbitrary Valuation Rings and Wild Ramification

Vaidehee Thatte

## Institution:

Binghamton University

## Time:

Thursday, April 29, 2021 - 3:00pm

## Location:

Zoom https://uci.zoom.us/j/98329625438

We aim to develop ramification theory for arbitrary valuation fields, extending the classical theory of complete discrete valuation fields with perfect residue fields. By studying wild ramification, we hope to understand the mysterious phenomenon of the defect (or ramification deficiency) unique to the positive residue characteristic case and is one of the main obstacles in obtaining resolution of singularities.

Extensions of degree p in residue characteristic p>0 are building blocks of the general case. We present a generalization of ramification invariants for such extensions. These results enable us to construct an upper ramification filtration of the absolute Galois group of Henselian valuation fields (joint with K.Kato).

# Generating series for counting finite nilpotent groups

## Institution:

Binghamton University

## Time:

Thursday, April 1, 2021 - 3:00pm to 4:00pm

## Location:

Zoom: https://uci.zoom.us/j/94971241077

Counting non-isomorphic finite nilpotent groups of order n is a very hard problem. One way to approach this problem is to count finite nilpotent groups of fixed nilpotency class c on d generators. The enumeration of such isomorphism classes of objects involves number theory and the theory of algebraic groups. However, very little is known about the explicit generating functions of these sequences of numbers when  c > 2 or d > 2. We use a direct enumeration of such groups that began in the works of M. Bacon, L. Kappe, et al, to provide a natural multivariable extension of the generating function counting such groups. Then we rederive the explicit formulas that are known so far.

# Gauss composition with level structure, polyharmonic Maass forms, and Hecke series

Olivia Beckwith

UIUC

## Time:

Thursday, April 8, 2021 - 3:00pm to 4:00pm

## Location:

Zoom: https://uci.zoom.us/j/95973703658

The Gauss composition law famously describes the class group of an order in a quadratic number field by an operation on binary quadratic forms up to matrix transformation. Using a stricter notion of equivalence, we describe ray class groups of a quadratic order in terms of quadratic forms.  We explore applications to representing primes by binary quadratic forms, and we describe leading coefficients of Hecke series for real quadratic fields as twisted traces of cycle integrals of polyharmonic Maass forms. This is ongoing joint work with Gene Kopp.

# Local-Global Phenomena for Elliptic Curves

Jacob Mayle

## Institution:

University of Illinois, Chicago

## Time:

Thursday, March 11, 2021 - 3:00pm to 4:00pm

## Location:

Zoom: https://uci.zoom.us/j/95528784206

A local-global principle is a result that allows us to deduce global information about an object from local information. A well-known example is the Hasse-Minkowski theorem, which asserts that a quadratic form represents a number if and only if it does so everywhere locally. In this talk, we'll discuss certain local-global principles in arithmetic geometry, highlighting two that are related to elliptic curves, one for torsion and one for isogenies. In contrast to the Hasse-Minkowski theorem, we'll see that these two results exhibit considerable rigidity in the sense that a failure of either of their corresponding everywhere local conditions must be rather significant.

# An isometric invariant of combinatorial type on (F_q^n,x_1^2+...+x_n^2) over F_q

Semin Yoo

## Institution:

University of Rochester

## Time:

Thursday, January 14, 2021 - 3:00pm to 4:00pm

## Location:

Zoom: https://uci.zoom.us/j/94525934294
In this talk, we introduce a new isometric invariant of combinatorial type on the quadratic space $(\mathbb{F}_{q}^{n},x_{1}^{2}+\cdots+x_{n}^{2})$ over $\mathbb{F}_{q}$. First, we recall some basic facts about quadratic forms. In particular, we will restrict ourselves to the case, where the base field is finite. In order to define this new invariant, we introduce special types of lines, named after line types in Minkowski's geometry. We prove that counting lines of each type is an isometric invariant on the quadratic space $(\mathbb{F}_{q}^{n},x_{1}^{2}+\cdots+x_{n}^{2})$ over $\mathbb{F}_{q}$. In order to prove this theorem, we redrive Minkowski's formula for the size of spheres on finite fields in an elementary way. Only some elementary facts of number theory are required for this talk.

# Computing modular forms using supersingular isogeny graphs

Alex Cowan

## Institution:

Harvard University

## Time:

Thursday, February 4, 2021 - 3:00pm to 4:00pm

## Location:

Zoom: https://uci.zoom.us/j/93364693756

We give an efficient algorithm for computing Fourier expansions of weight 2 cusp forms of prime level. The algorithm is based on Mestre's Methode des Graphes and supersingular isogeny graphs, and was used to greatly expand tables in the LMFDB. We'll also talk briefly about work in progress with Kimball Martin about heuristics for estimating the number of degree 2 cusp forms up to a given level.

# Smooth Cubic Surfaces with at Least 9 Lines

Fatma Karaoğlu

## Institution:

Tekirdağ Namık Kemal University

## Time:

Thursday, March 4, 2021 - 3:00pm to 4:00pm

## Location:

Zoom: https://uci.zoom.us/j/99322295399

A cubic surface is an algebraic variety of degree three in projective three space.  We will study cubic surfaces over different fields.  We are interested in the number of points and lines on a smooth cubic surface.  In this talk, we will focus on smooth cubic surfaces with at least 9 lines.  There are three cases with 27, 15 and 9 lines, respectively.  We will describe these surfaces in terms of normal forms, each of which involves either 4 or 6 parameters over the given field.  Using birational maps, the rational pooints on these normal forms will be described explicitly.