Department of Mathematical Sciences

309 SCEN

University of Arkansas

Fayetteville, AR 72701

P 479-575-3351

F 479-575-8630

E-mail: math@uark.edu

# Spring Lecture Series

**43**^{rd} Annual Spring Lecture Series

^{rd}Annual Spring Lecture Series

**Old and New themes in p-adic Cohomology**

**April 4-8, 2018 (Times TBA)**

Principle Speaker: Kiran Kedlaya (University of California at San Diego)

Public Lecture: Fernando Gouvêa (Colby)

### Associated Speakers

Bryden Cais (University of Arizona)

Richard Crew (University of Florida)

Veronika Ertl (University of Regensburg/Keio University)

David Hansen (Columbia University)

Ruochuan Liu (BICMR - Peking University)

Tong Liu (Purdue University)

Ananth Shankar (MIT)

Alberto Vezzani (LAGA - Université Paris 13)

Liang Xiao (University of Connecticut)

David Zureick-Brown (Emory University)

### Schedule of Events

Tentitive schedule of events as of February 8th.

April 4th: Graduate workshop (in the afternoon)

April 5th through the 7th: Reserved for main and contributing talks and public lecture (taking place the evening of the 5th)

April 8th: Travel day

**Speaker: Bryden Cais****Title:** "Breuil–Kisin modules via crystalline cohomology"**Abstract:** Let *k* be a perfect field of characteristic *p*, and *K* a totally ramified extension of *W(k)[1/p]*. The theory of Breuil-Kisin modules provides a classification of stable lattices
in crystalline *p*-adic representations of the absolute Galois group, *G _{K}*, of

*K*via finite-height Frobenius modules over the power series ring

*W(k)[[u]]*. On the other hand, the

*i*-th integral

*p*-adicétale cohomology of a smooth and proper formal scheme

*X*over the ring of integers

*O*in

_{K}*K*provides a stable lattice in a crystalline

*p*-adic

*G*-representation, and so has a Breuil-Kisin module attached to it. In this case, it is natural to ask if the associated Breuil- Kisin module can be described in terms of the crystalline cohomology of

_{K}*X*. Recent work of Bhatt, Morrow, and Scholze provides such a description after extending scalars to the period ring

*A*. In this talk, I will explain how to descend this result to obtain the Breuil-Kisin module over

_{inf}*W(k)[[u]]*when

*i < p − 1*and the crystalline cohomology of the special fiber of

*X*is

*p*-torsion-free in degrees

*i*and

*i*+ 1.

This is joint work with Tong Liu.

**Speaker: Richard Crew****Title:** "Isocrystals on a noetherian base"**Abstract:** Berthelot’s theory of overconvergent isocrystals provides an analogue, for smooth
schemes over a field of positive characteristic, of the notion of a local system.
As in the classical case this category is equivalent to the category of modules over
a suitable ring of differential operators. In this talk I will construct a generalization
of Berthelot’s rings of arithmetic differential operators, and use it to construct
a category of p-adic local systems on any scheme of finite type over a field of positive
characteristic.

**Speaker: Veronika Ertl****Title:** "Comparison of crystalline syntomic and rigid syntomic cohomology for strictly semistable
schemes."**Abstract:** We prove a comparison isomorphism between Nekovár and Niziol’s syntomic cohomology and log rigid syntomic cohomology for a strictly
semistable scheme with a nice compactification. Key points are a generalization of
Große-Klönne’s log rigid cohomology and the compatibility of crystalline and rigid
Hyodo-Kato maps on the Frobenius eigenspaces.

This is joint work with Kazuki Yamada.

**Speaker: David Hansen****Title:** "Zariski-constructible sheaves on rigid spaces"**Abstract:** Constructible sheaves play an important role in the classical theory ofétale cohomology.
In this talk, I’ll define an analogous class of so-called “Zariski-constructible”
étale sheaves on rigid analytic spaces, and explain some of their basic properties.
Along the way, I’ll discuss some new comparison theorems.

**Speaker: Kiran Kedlaya****Title:** "Old and new themes in p-adic cohomology"**Abstract:** One of the crowning achievements of mathematics in the 20th century is the resolution
of Weil’s conjectures on the number of rational points on algebraic varieties over
finite fields. These conjectures have consequences through algebraic number theory,
as well as in the theory of automorphic forms (the Ramanujan-Petersson conjecture),
cryptography (computation of orders of class groups of curves over finite fields,
which figure into modern public-key encryption and signature schemes), coding theory
(error-correcting codes based on algebraic geometry over finite fields), etc. In addition,
the methods developed for the proof themselves form the backbone of modern algebraic
geometry, with further applications too numerous to describe here.

Often unnoticed in this story is the fact that the original approach to the Weil conjectures,
via Grothendieck’s idea of lgebraizing the topological theory of covering spaces
(leading to the concept of *étale cohomology*), is not the only possible approach. An alternate possibility is to use methods originating
from Dwork’s proof of one of the Weil conjectures (rationality of the zeta function);
these were subsequently systematized into a theory of *p-adic cohomology* through the initial efforts of Berthelot and subsequent work of numerous authors.
This construction provides complementary information toétale cohomology in many theoretical
contexts; in addition, it provides important computational techniques unavailable
from étale cohomology, some of which are widely deployed in modern computer algebra
systems like **Magma** and **Sage**.

The backbone of *p*-adic cohomology is a form of de Rham cohomology for rigid analytic spaces; one may
then imagine that by analogy with the de Rham comparison theorem relating singular
and de Rham cohomology for C∞ manifolds (or the Dolbeaut version for complex manifolds),
one may be able to compareétale and de Rham cohomology for varieties over *p-*adic fields. This is true and forms the heart of *p-adic Hodge theory*. While many techniques are used in this subject, we will focus on the key concept
of a *perfectoid space*, which plays a central role in recent work on this topic by Scholze, Kedlaya-Liu,
Bhatt-Scholze-Morrow, et al. A perfectoid space is a certain type of nonarchimedean
analytic space (technically an *adic* *space* in the sense of Huber) which transforms canonically between characteristic 0 and characteristic *p*; this provides a powerful mechanism for systematically using tools from positive-characteristic
geometry (e.g., the Frobenius map) to make statements about geometry over *p*-adic fields.

The lecture series will consist of five standalone but related lectures, on the following

topics.

- Analytic and algebraic de Rham cohomology:

The cohomology of differential forms was originally introduced in the concept of real differentiable manifolds (de Rham), and later transferred to complex analytic varieties (Dolbeaut) and algebraic arieties (Atiyah-Hodge, Grothendieck). We illustrate this progression; in the process, we see how additional structures that appear originally at the analytic level (e.g., the Hodge filtration, the Gauss-Manin connection in a family) acquire algebraic interpretations (e.g., through the work of Katz-Oda). - Crystalline and rigid cohomology:

The properties of zeta functions over finite fields of characteristic p are usually studied using étale cohomology with coefficients in the field*Q*for some prime_{l}*l*≠*p*. However, this construction breaks down when*l*=*p*, and ideas from the theory of algebraic de Rham cohomology are needed to replace it. These ideas, developed initially by Berthelot, turn out to be closely related to the approach to zeta functions via*p*-adic analysis developed by Dwork. - Computational applications:

Since 2000, a variety of methods have been developed that apply the ideas and methods of*p*-adic cohomology to problems of computational number theory. These include the determination of zeta functions for varieties over finite fields and (via methods of Chabauty, Coleman, and Kim) rational points for varieties over number fields. - Perfectoid spaces:

Perfectoid spaces provide a consolidation of several key ideas in*p*-adic Hodge theory, such as the Fontaine-Wintenberger field of norms construction and the Faltings almost purity theorem. Moreover, several natural infinite towers of spaces admit inverse limits in the category of perfectoid spaces, giving rise to exotic but important new geometric structures. - Applications of perfectoid spaces in p-adic Hodge theory:

We describe in detail how perfectoid spaces can be used to clarify and extend several key results in*p*-adic Hodge theory, such as the explicit description of étale fundamental groups of analytic spaces, and the comparison isomorphism between étale and deRham cohomology for smooth proper spaces (without and with coefficients).

**Speaker: Brian Lawrence****Title:** "Diophantine problems and a *p*-adic period Map"**Abstract:** We discuss a proof of the *S*-unit theorem and Faltings’ theorem (finiteness of

rational points on higher-genus curves) using *p*-adic Hodge theory and a *p*-adic period map.

**Speaker: Ruochuan Liu****Title:** "Towards a *p*-adic Riemann-Hilbert correspondence"**Abstract:** I’ll report the recent progress on the p-adic version of Riemann-Hilbert correspondence.

Joint work with Xinwen Zhu.

**Speaker: Tong Liu****Title:** "Fontaine-Messing theory over power series ring"**Abstract:** Let *k* be a perfect field of characteristic *p* > 2, *R = W(k)[[t _{1}, . . . , t_{d}]]* a power series ring and

*X*a proper smooth scheme over

*R*. I will discuss an ongoing project with Yong Suk Moon and Deepam Patel. Our goal is to extend Fontaine-Messing’s theory form

*W(k)*to

*R*, that is, to establish comparison between torsion crystalline cohomology and torsionétale cohomology of

*X*.

**Speaker: Yong Suk Moon****Title:** "Relative crystalline representations and *p*-divisible groups"**Abstract:** Let k be a perfect field of characteristic *p* > 2, and let *K* be a finite totally ramified extension over *W(k)[ 1/**p**]*. Kisin showed that any crystalline Gal(^{over bar}*K/K*) representation whose Hodge-Tate weights lie in [0, 1] arises from a *p*-divisible group over *O _{K}*. We study the analogous problem for relative crystalline representations over the
unramified base ring

*R = W(k)[[t*, and identify the relative crystalline representations which arise from a

_{1}, ..., t_{d}]]*p*-divisible group over

*R*.

**Speaker: Ananth Shankar****Title:** "The p-curvature conjecture and families of varieties."**Abstract:** The Grothendieck-Katz *p*-curvature conjecture is an analogue of the Hasse principle for differential equations.
It states that a set of arithmetic differential equations on a variety has finite
monodromy if its *p*-curvature vanishes modulo *p*, for almost all primes *p*. We will discuss the case of families of varieties, and among other things, will
prove that every simple closed loop has finite monodromy if the variety is a generic
family of curves.

**Speaker: Alberto Vezzani****Title:** "Cohomological realizations of rigid analytic motives"**Abstract:** We present the homotopical framework of motives applied to rigid analytic varieties
and adic spaces. We show how it can be used to generalize some properties of classical
cohomology theories and, when intertwined with the tilting equivalence, even to define
new interesting ones.

**Speaker: Liang Xiao****Title:** "Slopes of modular forms and ghost conjecture of Bergdall and Pollack"**Abstract:** For a *p*-adic modular form that is an eigenvalue of the *U _{p}*-operator, its slope is the

*p*-adic valuations of the

*U*eigenvalue. The slopes of modular forms were studied by Coleman, Gouvêa, Mazur, Buzzard, Calegari, and many mathematicians. Recently, Bergdall and Pollack formulated a conjecture that gives the precise slopes of all

_{p}*p*-adic modular forms under the so-called Buzzard-regular condition. We give a reformulation in purely representation theory of GL

_{2}(

**Q**

*), and explain its relation to*

_{p}*p*-adic local Langlands program.

**Speaker: David Zureick-Brown****Title:** "*p*-adic cohomology and le Stum’s Overconvergent Site"**Abstract:** Classically, the construction of rigid cohomology is a bit complicated and

requires many choices. A recent advance is the construction by le Stum of an ‘Overconvergent

site’ which computes the rigid cohomology of a variety *X*. This site involves no choices and

so it is trivially well defined, and many things (like functoriality) become transparent.
I’ll

explain this construction and several applications.