# University of Chicago Number Theory Seminar

### SYNOPSIS

The UChicago Number Theory seminar will meet on Thursday afternoons, from 2:00 to 3:30.

We ask speakers to divide talks into two sections. The first section should be directed at a non-expert audience (first- and second-year students from a different area of number theory), and should last about half an hour.

We ask students, especially non-expert students, to be aggressive in asking questions.

See here .

### SCHEDULE

Thursdays, 2:00–3:30 PM, Eckhart Hall, Room 117, unless noted otherwise

 FALL 2019 Oct. 3, 2019 Junho Peter Whang (MIT) Oct. 10, 2019 Arthur-César Le Bras (Paris XIII) Oct. 17, 2019 Sean Howe (Utah) Oct. 24, 2019 Preston Wake (Michigan State) Oct. 31, 2019 Simon Marshall (Wisconsin) Nov. 7, 2019 Kai-Wen Lan (Minnesota) Nov. 14, 2019 Andrea Dotto (UChicago) Nov. 21, 2019 Shai Evra (IAS) Nov. 28, 2019 No seminar Dec. 5, 2019 Dimitris Koukoulopoulos (Montreal) Dec. 12, 2019 Ananth Shankar (MIT)

### ABSTRACTS

#### Oct. 3, 2019 Thu, 2:00–3:30 PM, Eckhart 117.

Junho Peter Whang, MIT

Diophantine analysis on moduli of local systems

Moduli spaces for special linear rank two local systems (with prescribed boundary traces) on topological surfaces are basic objects in geometry. After motivating their Diophantine study, we use mapping class group dynamics and differential geometric tools (such as harmonic maps and systolic inequality) to establish a structure theorem for the integral points of these varieties. This invites an analogy with classical finite generation results on log Calabi-Yau varieties of linear type, such as algebraic tori and abelian varieties, and motivates the Diophantine analysis of subvarieties on the moduli spaces. In this direction, we give an effective analysis of integral points for nondegenerate algebraic curves. Time permitting, we also discuss more recent work on this and related problems.

#### Oct. 10, 2019 Thu, 2:00–3:30 PM, Eckhart 117.

Arthur-César Le Bras, Paris XIII

Prismatic Dieudonné theory

I will discuss joint work with Anschütz, which establishes classification theorems for $p$-divisible groups, using the recent prismatic formalism of Bhatt-Scholze.

#### Oct. 17, 2019 Thu, 2:00–3:30 PM, Eckhart 117.

Sean Howe, Utah

p-Adic modular forms and a completed Kirillov model

The space of p-adic modular forms was first constructed by Serre as a completion of the space of classical modular forms for the p-adic topology on q-expansions. This completed space of p-adic modular forms exhibited new structures not present in classical modular forms: for example, p-adic weights and a differential operator theta. Subsequent work of Katz showed that p-adic modular forms are functions on a profinite cover of the ordinary locus and gave a natural geometric origin for $p$-adic weights and the operator theta; the importance of these ideas was further demonstrated by work of Hida constructing weight families of (ordinary) p-adic modular forms. In this talk, we explain how p-adic weights, the theta operator, and ordinary forms all have a unified representation-theoretic origin by passing to an even deeper cover of the ordinary locus: the (big) Igusa variety of Caraiani-Scholze. This Igusa variety admits a moduli-theoretic action of a (very big) group that subsumes the Z_p^\times action of Katz's theory and the differential operator theta, and Serre's construction of p-adic modular forms via completion of $q$-expansions extends to a construction of functions on the big Igusa variety as a p-adic completion of (twists of) the Kirillov models of the automorphic representations attached to classical modular forms.

#### Oct. 24, 2019 Thu, 2:00–3:30 PM, Eckhart 117.

Preston Wake, Michigan State

Tame derivatives and the Eisenstein ideal

As was made famous by Mazur, the mod-5 Galois representation associated to the elliptic curve X_0(11) is reducible. Less famously, but also noted by Mazur, the mod-25 Galois representation is reducible. We’ll talk about this kind of extra reducibility phenomenon more generally, for cuspforms of even weight k and prime level. We'll observe that the characters appearing in the reducible representation are related, on one hand, to an algebraic invariant (the ‘tame deriviative’ of an L-function), and, on the other hand, to an algebraic invariant (the 'tame L-invariant'). This type of 'algebraic=analytic' relation is predicted by a version of the Bloch-Kato conjecture for families of motives formulated by Kato.

#### Oct. 31, 2019 Thu, 2:00–3:30 PM, Eckhart 117.

Simon Marshall, Wisconsin

TBA

TBA

#### Nov. 7, 2019 Thu, 2:00–3:30 PM, Eckhart 117.

Kai-Wen Lan, Minnesota

TBA

TBA

#### Nov. 14, 2019 Thu, 2:00–3:30 PM, Eckhart 117.

Andrea Dotto, UChicago

TBA

TBA

Shai Evra, IAS

TBA

TBA

#### Dec. 5, 2019 Thu, 2:00–3:30 PM, Eckhart 117.

Dimitris Koukoulopoulos, Montreal

TBA

TBA

#### Dec. 12, 2019 Thu, 2:00–3:30 PM, Eckhart 117.

Ananth Shankar, MIT

TBA

TBA