Arithmetic Geometry and Number Theory Group
at
Università degli Studi di Milano

Title: On Tangency in Log Geometry

Abstract: Notions of "infinitesimal information" are a recurring theme in mathematics. With its great flexibility, Lurie's cotangent complex formalism goes far in unifying this theme. There is no established notion of quasi-coherent sheaves on a log scheme. As such, the relevant constructions in log geometry do not immediately fit in Lurie's framework. I will explain how infinitesimal constructions in log geometry (e.g., log derivations and the log cotangent complex) can be expressed purely in categories that do not see the log structure and fit in Lurie's formalism. We will then reap the rewards: a well-behaved deformation theory for animated log rings and log ring spectra and a rigidity result for log étale extensions.

Title: Syntomic cohomology and real topological cyclic homology

Abstract: In this talk, I will show that real topological cyclic homology admits a complete exhaustive filtration whose graded pieces are equivariant suspensions of syntomic cohomology. Combined with the announced results of Antieau-Krause-Nikolaus and Harpaz-Nikolaus-Shah, this would lead to the computation of the equivariant slices of the real K-theory of Z/p^n after a certain suspension. The key ingredients of the proof are a real refinement of the Hochschild-Kostant-Rosenberg filtration and the computation of real topological Hochschild homology of perfectoid rings in my joint work with Hornbostel.

The detailed schedule is available here.

Title: Plectic insights on the BSD conjecture for higher rank elliptic curves

Abstract: Heegner points play a pivotal role in our understanding of the arithmetic of modular elliptic curves. They arise from CM points on Shimura curves and control the Mordell-Weil groups of elliptic curves of rank 1. The work of Bertolini, Darmon and their school has shown that \(p\)-adic methods can be successfully employed to generalize the definition of Heegner points to quadratic extensions that are not necessarily CM. Notably, Guitart, Masdeu, Sengun have defined and numerically computed Stark-Heegner points in great generality. Their computations strongly support the belief that Stark-Heegner points completely control the Mordell-Weil groups of elliptic curves of rank 1.
  In this talk we will survey plectic generalizations of Stark-Heegner points developed in a series of articles with Darmon, Gehrmann, Guitart and Masdeu. These plectic Stark-Heegner points were inspired by Nekovar-Scholl’s plectic conjectures and should help illuminate the arithmetic intricacies of higher rank elliptic curves. Their construction is \(p\)-adic, cohomological, and unfortunately lacking a satisfying geometric interpretation. Nevertheless, we formulated precise conjectures describing their arithmetic significance which we were able to substantiate with both numerical and theoretical evidence.

Title: Standard conjectures in Arakelov geometry: from the projective space to Grassmannians

Abstract: The standard conjectures, formulated by Grothendieck fifty-five years ago, remain one of the biggest challenges in the study of algebraic cycles. Moving from algebraic to arithmetic geometry, Gillet and Soulé have formulated analogues of the standard conjectures for varieties defined over the integers. Since the degrees of arithmetic cycles correspond to heights, these new standard conjectures can be used to obtain several number theoretical results, such as an effective version of the Bogomolov conjecture, as shown by Zhang. In this talk, based on joint work with Paolo Dolce and Roberto Gualdi, I will explain how one can study these standard conjectures for arithmetic varieties which admit a cellular decomposition. In particular, I will explain how to classify all the arithmetic line bundles on the projective that satisfy the standard conjectures, and I will outline a program to generalize this proof to Grassmannians, which uses a new presentation of their Arakelov-Chow ring.

Title: Horizontal \(p\)-adic \(L\)-functions and non-vanishing of twists of \(L\)-functions by characters of fixed order

Abstract: Given a positive integer \(d\) greater than or equal to 2, a fundamental question is to quantify how many order \(d\) character twists of a central \(L\)-value (or its derivatives) of a fixed modular form are non-vanishing. The question of \(d = 2\) falls under the purview of Goldfeld's conjecture, where substantial progress has been made in recent years using both analytic number-theoretic and Iwasawa-theoretic techniques. In the case of \(d\) greater than 2, conjectures of David-Fearnley-Kisilevsky predict that 100% of order d twists should be non-vanishing. However, little was previously known toward this conjecture as it lies beyond the current scope of analytic techniques. In this talk I will describe a new approach to studying the above questions using horizontal (i.e. prime-to-\(p\)) Iwasawa theory, via a new construction called horizontal \(p\)-adic \(L\)-functions. The non-vanishing of these horizontal \(p\)-adic \(L\)-functions is related to Kolyvagin's conjecture and other questions in the circle of ideas surrounding Euler systems. Using horizontal \(p\)-adic \(L\)-functions, we give strong quantitative lower bounds on the number of non-vanishing order d twists of the central \(L\)-value of a holomorphic newform as well as its derivative, and more generally on the simultaneous non-vanishing of such values for finitely many holomorphic newforms. For 100% of elliptic curves, we improve the previously best-known lower bounds in the \(d = 2\) case due to Ono and Perelli-Pomykala, and for \(d > 2\) we give the first general results toward David-Fearnley-Kisilevsky's conjecture.
This is joint work with Asbjørn Nordentoft.

Title: Rigid meromorphic cocycles for orthogonal groups

Abstract: I will talk about a generalization of Darmon and Vonk's notion of rigid meromorphic cocycles to the setting of orthogonal groups. These objects should be viewed as \(p\)-adic analogues of the meromorphic functions on orthogonal Shimura varieties with prescribed divisors constructed by Borcherds. After giving an overview over the general setting I will discuss the case of orthogonal groups attached to quadratic spaces of dimension 4 in more detail. In particular, I will highlight the similarities with the classical theory of Hilbert modular surfaces. This is an account of joint works with Henri Darmon and Michael Lipnowski, and with Xavier Guitart and Marc Masdeu.

Title: The role of primes of good reduction in the Brauer-Manin obstruction

Abstract: A way to study rational points on a variety is by looking at their image in the \(p\)-adic points. Some natural questions that arise are the following: are rational points dense in the \(p\)-adic points? If not, where does this obstruction come from? I will explain how primes of good reduction can play a role in the Brauer-Manin obstruction to weak approximation (i.e. the density of rational points in the \(p\)-adic points), with particular emphasis on the case of \(K3\) surfaces. I will then explain how the reduction type (ordinary or non-ordinary good reduction) plays a role.

Title: A \(p\)-adic explicit reciprocity law for diagonal classes

Abstract: Theorems known as reciprocity laws are ubiquitous in number theory. In this talk I will discuss a particular instance of \(p\)-adic reciprocity law that shall appear in my PhD thesis. This explicit reciprocity law relates certain diagonal classes on a triple product of modular curves to \(p\)-adic special values of a suitable \(p\)-adic \(L\)-function, extending work of Darmon-Rotger and Bertolini-Seveso-Venerucci. I will present this result and, time permitting, I will try to explain how it can be applied to shed some light on certain cases of the Birch and Swinnerton-Dyer conjecture.

Title: Infinitesimal rational actions

Abstract: For any \(k\)-group scheme of finite type \(G\), if there exists a generically free rational \(G\)-action on a \(k\)-variety \(X\), then the dimension of \(\operatorname{Lie}(G)\) is upper bounded by the dimension of the variety. During my PhD, I showed that this is the only obstruction to the existence of such actions, when \(k\) is a perfect field of positive characteristic and \(G\) is infinitesimal commutative trigonalizable. During my talk, I will give the motivation for this problem and explain the result in the case of the \(p\)-torsion of a supersingular elliptic curve.

Title: Conjectures on \(L\)-functions for varieties over function fields and their realisation

Abstract: (Joint work with T. Keller (Groningen) and Y. Qin (Regensburg)) We consider versions for smooth varieties \(X\) over finitely generated fields \(K\) in positive characteristic \(p\) of several conjectures that can be traced back to Tate, and study their interdependence. In particular, let \(A/K\) be an abelian variety. Assuming resolutions of singularities in positive characteristic, I will explain how to relate the BSD-rank conjecture for \(A\) to the finiteness of the \(p\)-primary part of the Tate-Shafarevich group of \(A\) using rigid cohomology. Furthermore, I will discuss what is needed for a generalisation.

Title: Period Relations for Arithmetic Automorphic Periods on Unitary Groups

Abstract: Given an automorphic representation of a unitary group, one can define an arithmetic automorphic period as the Petersson inner product of a deRham rational form. Here the deRham rational structure comes from the cohomology of Shimura varieties. When the form is holomorphic, the period can be related to special values of \(L\)-functions and is better understood. In this talk, we formulate a conjecture on relations among general arithmetic periods of representations in the same \(L\)-packet and explain a conditional proof.

Title: \(p\)-adic height pairings of geodesics

Abstract: We will discuss recent progress on a tentative theory of differences of singular moduli for real quadratic fields, and in particular how it leads to the construction of a mysterious p-adic height pairing of real quadratic geodesics on modular curves.

Title: Syntomic cohomology of real topological K-theory

Abstract: Work of Hahn—Raksit—Wilson extended the Bhatt—Morrow—Scholze filtration on topological cyclic homology and topological periodic cyclic homology to sufficiently nice commutative ring spectra. This allows one to define syntomic cohomology and prismatic cohomology at this level of generality. One example of such a nice commutative ring spectrum is the spectrum ko known as connective real topological K-theory. In joint work with Christian Ausoni and John Rognes, we compute syntomic cohomology of ko modulo \((2,\eta,v_1)\). As applications, we compute the topological cyclic homology and algebraic K-theory of ko modulo \((2,\eta,v_1)\). I will also mention applications to the Lichtenbaum—Quillen property and the telescope conjecture for algebraic K-theory and topological cyclic homology of ko.

Title: On two methods for \(p\)-adic \(L\)-functions

Abstract: \(p\)-adic \(L\)-functions are rigid analytic functions that interpolate complex \(L\)-values. As such, they encode similar local information as complex L-functions; the advantage is that they may be easier to relate to global arithmetic invariants, and that they admit extra variables of \(p\)-adic deformation. Constructing \(p\)-adic \(L\)-functions, however, is nontrivial. I will talk about two complementary ideas in this craft. The first one is to interpolate ratios of global and local zeta integrals; for the latter, one can use the local Langlands correspondence in families. The second one, in joint work with Wei Zhang, is to interpolate the geometric side of a relative-trace formula, in order to obtain \(p\)-adic \(L\)-functions from the spectral side.

Title: Non-vanishing of \(L\)-values for quadratic twists of elliptic curves

Abstract: We prove the non-vanishing of central \(L\)-values for families of quadratic twists of the elliptic curves with complex multiplication introduced by B. Gross in his thesis. From this, we obtain the finiteness of their Tate–Shafarevich group. This is joint work with Yong-Xiong Li.

Title: The integral Hodge polygon for Barsotti-Tate groups with endomorphism structure

Abstract: Let \(p\) be a prime number. A classical invariant of Barsotti-Tate groups (and more general \(p\)-adic Hodge-theoretic objects) is known as the Hodge polygon. For objects endowed with additional structure, such as the action of a finite extension of \(Z_p\) (endomorphism structure), this invariant can be refined in several ways. In this talk I will introduce the "integral Hodge polygon", a refinement of the Hodge polygon which is well suited for families of objects over a \(p\)-adic analytic space, as for instance the \(p\)-adic completion of a Shimura variety. This is joint work with Stéphane Bijakowski.

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.

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.