Arithmetic Geometry Seminar / Seminario di Geometria Aritmetica

**Title:** Non-commutative nature of \(l\)-adic vanishing cycles

**Abstract:** It is well known that the theory of vanishing cohomology is strictly related to that of singularity categories.
Indeed, in characteristic zero it is known after A. Efimov how to recover the vanishing cohomology with its monodromy action from the singularity category of the special fiber.
More recently, A. Blanc, M. Robalo, B. Toën and G. Vezzosi defined \(l\)-adic cohomology for non-commutative spaces (a.k.a. dg categories).
Moreover, they identified the \(l\)-adic cohomology of the singularity category of the special fiber of a scheme over an henselian trait with the (homotopy) fixed points of vanishing cohomology with respect
to the action of the inertia group. This holds true also in positive and mixed characteristics.
In this talk, I will explain how to recover the whole vanishing cohomology, together with the natural action of the inertia group. This is joint work with D. Beraldo.

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**Title:** The Frobenius height of prismatic cohomology group

**Abstract:** Introduced by Deligne, cohomology group of a complex algebraic variety underlies the notion of mixed Hodge structure. Moreover, such a notion can be extended to cohomology of general coefficients, including variations of mixed Hodge structures. In this talk, we consider the analogue in integral \(p\)-adic geometry: the prismatic cohomology and its structure of prismatic \(F\)-crystal. In particular, we give a bound of the Frobenius eigenvalues of the prismatic cohomology with coefficients, analogous to the bound of weights of cohomology in complex geometry. This is a joint work in progress with Shizhang Li.

**Title:** Constructible sheaves on schemes and a categorical Künneth formula

**Abstract:** I will present a uniform theory of constructible and lisse sheaves, with coefficients in a general condensed coefficient ring, for arbitrary schemes. Among other things, this recovers and extends the existing approaches to \(\ell\)-adic constructible sheaves in the literature. In the second part of the talk, I will talk about a categorical Künneth formula for Weil sheaves. This is joint work with Tamir Hemo and Timo Richarz.

**Room:** Aula C03, Via Mangiagalli 25, Facoltà di Scienze Agrarie e Alimentari

**Speakers:** Andrea Bandini (Università degli Studi di Pisa), Francesco Sala (Università degli Studi di Pisa), Tamás Szamuely (Università degli Studi di Pisa), Angelo Vistoli (Scuola Normale Superiore, Pisa).

The detailed schedule is available here.

**Title:** Rational \(p\)-adic Hodge theory for rigid-analytic varieties

**Abstract:** In this talk, I will discuss the rational \(p\)-adic Hodge theory of general \(p\)-adic rigid-analytic varieties, without properness assumptions. The study of this subject for varieties that are not necessarily proper (e.g. Stein) is motivated in part by the desire of finding a geometric incarnation of the \(p\)-adic Langlands correspondence in the cohomology of local Shimura varieties. In this context, one difficulty is that the relevant cohomology groups (such as the \(p\)-adic (pro-)étale, and de Rham ones) are usually infinite-dimensional, and, to study them, it becomes important to exploit the topological structure that they carry. But, in doing so, one quickly runs into several topological issues: for example, the category of topological abelian groups is not abelian, and the cohomology groups of a complex of topological vector spaces can be pathological in the case the differentials do not have closed image. I will explain how to overcome these issues, using the condensed and solid formalisms recently developed by Clausen and Scholze, and I will report on a general comparison theorem describing the geometric rational \(p\)-adic (pro-)étale cohomology in terms of de Rham data.

**Title:** Motivic action conjectures

**Abstract:** A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. A recent series of conjectures proposes an arithmetic explanation: a hidden degree-shifting action of a higher Chow group (motivic cohomology group). We will give an overview of these conjectures, focusing on the examples of \(GL(2)\) over \(Q\) and over quadratic fields, and \(GSp(4)\) over \(Q\).

**Title:** Compactifications of moduli spaces and stratified homotopy theory

**Abstract:** Compactifications of locally symmetric spaces or more generally moduli spaces often come equipped with natural stratifications, that is, a ''well-behaved'' partition of the space. Concrete examples include the Borel-Serre and reductive Borel-Serre compactifications of the locally symmetric space associated to an arithmetic group, and the Deligne-Mumford-Knudsen compactification of the moduli stack of stable curves. Arising from this additional structure are a wealth of interesting constructible (complexes of) sheaves, i.e. sheaves which are locally constant along each stratum (but not necessarily on the whole space!). These in turn define interesting cohomology theories, e.g. intersection cohomology and weighted cohomology.

It is a classical result that locally constant sheaves on a sufficiently nice topological space are classified by the fundamental groupoid, or homotopy type. For stratified spaces, we have a similar classification of constructible sheaves as representations of the so-called exit path category, or stratified homotopy type. Calculating the stratified homotopy type of a concrete stratified space would allow us to study the constructible sheaves from a more combinatorial viewpoint - in theory at least.

I will talk about some explicit calculations.

**Title:** Brauer groups and Neron-Severi groups of surfaces over finite fields

**Abstract:** For a smooth and proper surface over a finite field, the formula of Artin and Tate relates the behaviour of the zeta-function at \(1\) to other invariants of the surface. We give a version of the formula which equates invariants related to the Brauer group to invariants to the Neron-Severi group. To illustrate our results we give some applications for abelian surfaces.

**Title:** On the left Kan extension of the Chow group of zero cycles

**Abstract:** We explain different types of Chow groups of zero cycles for singular schemes and how they
should be related. More precisely, we show that the Levine-Weibel Chow group of zero cycles is left Kan
extended from smooth algebras in the affine case over algebraically closed fields, and even rigid for
surfaces. One key ingredient is Bloch's formula for singular schemes. The motivation for these results comes
from algebraic \(K\)-theory. This is joint work in progress with Matthew Morrow.

**Title:** Generalized Weibel’s conjecture

**Abstract:** This is a report of a joint work with Shane Kelly and Georg Tamme.

The main result affirms that for a qcqs derived scheme \(X\) whose underlying scheme has finite valuative dimension \(d\), we have \(K_i(X)=0\) for \(i<-d\). If \(X\) is a noetherian scheme, the result is due to Kerz-Strunk-Tamme. The method of the proof follows Kerz-Strunk-Tamme who deduced it from the pro-cdh descent for algebraic \(K\)-theory. The latter property fails for non-noetherian schemes in general. A key point is that it still holds replacing schemes by derived schemes.

**Title:** A motivic integral p-adic cohomology

**Abstract:** We use the theory of logarithmic motives to construct an integral \(p\)-adic cohomology
theory for smooth varieties over a field \(k\) of characteristic \(p\), that factors through the category of
Voevodsky (effective) motives. If \(k\) satisfies resolutions of singularities, we will show that it is indeed
a “good" integral \(p\)-adic cohomology and it agrees to a similar one constructed by Ertl, Shiho and Sprang:
we will then deduce many interesting motivic properties.

**Title:** A stacky perspective on p-adic non-abelian Hodge theory

**Abstract:** \(p\)-adic non abelian Hodge theory, also known as the \(p\)-adic Simpson correspondence, aims at describing \(p\)-adic local systems on a smooth rigid analytic variety in terms of Higgs bundles. I will explain in this talk why the « Hodge-Tate stacks » recently introduced by Bhatt-Lurie and Drinfeld in their work on prismatic cohomology can be useful to study this kind of questions. Joint work with Johannes Anschütz and Ben Heuer.

**Title:** On the Tate conjecture for divisors

**Abstract:** We prove that the Tate conjecture in codimension \(1\) over a finitely generated field
follows from the same conjecture for surfaces over its prime subfield. In positive characteristic, this is due
to de Jong-Morrow over \(\mathbf{F}_p\) and to Ambrosi for the reduction to \(\mathbf{F}_p\). We give a
different proof than Ambrosi's, which also works in characteristic \(0\); over \(\mathbf{Q}\), the reduction
to surfaces follows from a simple argument using Lefschetz's \((1,1)\) theorem.

**Title:** The Green-Tao theorem for number fields (and beyond)

**Abstract:** Green and Tao famously proved that there are arbitrarily long arithmetic progressions of
prime numbers. Around the same time, Tao proved an analogue for the Gaussian integers: the set of prime
elements of \(\mathbb{Z}[i]\) contains constellations of arbitrary shapes. After reviewing some background of
these theorems, I will explain our generalization of them to the context of prime elements in general number
fields, a joint result with M. Mimura, A. Munemasa, S. Seki and K. Yoshino.
Time permitting, I will describe my recent attempt to deepen this result, following the work of
Green-Tao-Ziegler on more complex linear patterns of prime numbers.

**Title:** A p-adic 6-Functor Formalism in Rigid-Analytic Geometry

**Abstract:** We introduce a p-adic 6-functor formalism on rigid varieties and more generally Scholze's
diamonds, which in particular proves Poincaré duality for étale \(\mathbb F_p\)-cohomology on proper smooth
rigid varieties over mixed-characteristic fields. The basic idea is to employ Clausen-Scholze's condensed
mathematics in order to construct a category of "quasicoherent complete topological \(\mathcal
O^{+a}_X/p\)"-sheaves on any diamond X. This category satisfies v-descent and admits the usual six functors
\(\otimes\), \(\underline{Hom}\), \(f^*\), \(f_*\), \(f_!\) and \(f^!\) with all the expected compatibilities.
One can then pass to the category of \(\varphi\)-modules, i.e. pairs \((M, \varphi_M)\) where \(M\) is as
before and \(\varphi_M\colon M \to M\) is a Frobenius-semilinear isomorphism. By proving a version of the
\(p\)-torsion Riemann-Hilbert correspondence we show that classical étale \(\mathbb F_p\)-sheaves embed fully
faithfully into the category of \(\varphi\)-modules (identifying perfect sheaves on both sides), which finally
allows us to relate the 6-functor formalism of \(\varphi\)-modules to \(\mathbb F_p\)-cohomology. With this
theory established, we also obtain a new and short proof of the primitive comparison isomorphism.

**Title:** Real quadratic fields with large class number

**Abstract:** We know that every integer can be factored in a unique way as a product of primes. This is no
longer true over number fields and the class number indicates "how badly unique factorization fails": if the
class number is one then we have unique factorization, while anything bigger than one means we don’t. A
long-standing open conjecture of Gauss states that there are infinitely many real quadratic fields with class
number one. In the opposite direction, one can prove that there are infinitely many real quadratic fields with
class number as large as possible. In this talk I will explain what ”as large as possible” means and a few
ideas on how the result can be proved. This is joint work with Fazzari, Granville, Kala and Yatsyna.

**Title:** On the first derivative of cyclotomic Katz p-adic L-functions at exceptional zeros.

**Abstract:** This talk is based on a joint work with Ming-Lun Hsieh studying the exceptional zeros
conjecture of Katz \(p\)-adic \(L\)-functions. We will present a formula relating the first derivative of the
cyclotomic Katz \(p\)-adic L-function attached to a ring class character of a general CM field to the product
of an \(L\)-invariant and the value of some improved Katz p-adic L-function at \(s=0\). In particular, we show
that these Katz \(p\)-adic \(L\)-functions have a simple trivial zero if and only if their cyclotomic
\(L\)-invariants are non-zero. Our method uses congruences of Hilbert CM forms and the theory of deformations
of reducible Galois representations. I will discuss at the end of this talk about how we can compute the first
derivative beyond the case where the branch character is a ring class character using \(p\)-adic Eisenstein
congruences for \(U(2,1)\).

**Title:** Homotopical methods for Hyodo-Kato cohomologies.

**Abstract:** Using homotopical methods in rigid analytic geometry, we show how to give a streamlined
definition of the Hyodo-Kato cohomology for rigid analytic varieties over a non-archimedean local field with
residue characteristic \(p>0\). As an application, we deduce an exact complex à la Clemens-Schmidt involving
the monodromy operator, the rigid and the \(log\)-rigid cohomologies. Work in progress with F. Binda and M.
Gallauer.

**Title:** Hida theory for Special quaternionic orders.

**Abstract:** In this talk, we discuss a quaternionic Control Theorem, in the spirit of Hida and
Greenberg-Stevens, considering a generalization of Eichler orders proposed by Pizer. These orders allow higher
level-structure at the primes where the quaternion algebra ramifies. Interestingly, the quaternionic modular
forms associated with these orders live in Hecke-eigenspaces whose rank might be 2 and not necessarily 1, as
in the Eichler case. The proven Control Theorem deals with this higher multiplicity situation. Time
permitting, we will discuss some work-in-progress developments on recovering strong multiplicity 1, and an
expected generalization of Chenevier's \(p\)-adic extension of the Jacquet-Langlands correspondence with these
interesting level structures. This last part is joint work with Aleksander Horawa.

**Title:** On a reciprocity law for \(GSp(4)\) and arithmetic applications.

**Abstract:** After briefly discussing \(p\)-adic type Birch and Swinnerton-Dyer conjectures, I will
explain a reciprocity law supporting it which is a work in collaboration with Fabrizio Andreatta, Massimo
Bertolini and Rodolfo Venerucci.

**Title:** Classification of number fields with small regulator.

**Abstract:** Classification of number fields with bounded invariants is an important problem in
Computational Algebraic Number Theory. In this talk, we shall focus on a procedure which classifies number
fields with small regulator: in particular, we show that better results are available if a specific function,
which is a key object in the procedure, is estimated as sharply as possible. As a consequence, we rigorously
provide minimum regulators for number fields of degree 8 and with 1 complex place.