AGNT Ical Atom

Jacobians of CM curves are abelian varieties with a particularly large endomorphism algebra, which provides them with a rich arithmetic structure. The motivating question for the results in this talk is whether we can find hyperelliptic and non-hyperelliptic curves with maximal CM by a given order whose Jacobians are isogenous. Joint work with Sorina Ionica, and Jeroen Sijsling considers this question in genus 3 by using the catalogue of CM fields in the LMFDB, and found a (small) list of such isogenous Jacobians. This talk describes the main constructions, some results, and Shimura class groups.

In the last twenty years algebraic geometry has evolved rapidly, from the geometry of schemes and stacks, to the derived algebraic geometry (DAG) of today. The flavor of contemporary DAG is very homotopical, in the sense that is largely based on simplicial sets and Quillen model structures.

This talk is on another approach to DAG, of a very algebraic flavor, which avoids simplicial methods and model structures altogether. Instead, the fundamental concept is that of DG rings, traditionally called unital associative cochain DG algebras. DG rings are of two distinct kinds: noncommutative and commutative. These two kinds of DG rings interact, primarily through central DG ring homomorphisms; and this interaction is quite fruitful. The main tool for studying DG rings, DG modules over them, and the associated derived categories, is the construction and manipulation of resolutions. Hence “constructive approach”.

I will define the notions mentioned above, and state several results, among them: (1) The squaring operation and Van den Bergh’s rigid dualizing complexes in the commutative arithmetic setting; (2) Theorems on derived Morita theory; (3) Duality and tilting for commutative DG rings. I will try to demonstrate that this constructive approach is very amenable to calculation. I will also mention work of Shaul, within this framework, on derived completion of DG rings and on the derived CM property.

The talk will conclude with a couple of conjectural ideas: (a) A structural description of the derived category of DG categories; (b) A construction of the cotangent DG module within the framework of commutative DG rings, without any arithmetic restrictions.

Some of this work is joint with J. Zhang, L. Shaul, M. Ornaghi and S. Singh.

Slides for the talk are available here:

https://sites.google.com/view/amyekut-math-bgu/home/lectures/constr-der-algebra

(updated 15 March 2022)

Classically, Diophantine approximation deals with the problem of studying “good” approximations of a real number by rational numbers. I will explain the meaning of “good approximants” and the classical main results in this area of research. In particular, Klaus Roth was awarded with the Fields medal in 1955 for proving that the approximation exponent of a real algebraic number is 2. I will present a recent extension of Roth’s theorem in the framework of adelic curves. These mathematical objects, introduced by Chen and Moriwaki in 2020, stand as a generalisation of global fields.

Ribet’s lemma is an algebraic statement that Ribet used in his proof of the converse of Herbrand’s theorem. Since then various generalisations of Ribet’s lemma have been found, with arithmetic applications. In this talk I will discuss a joint work with Ariel Weiss in which we show that two measures of reducibility for two dimensional representations over a DVR are the same, thus answering a question of Bellaiche and Cheneveier, and deducing from it a particular generalisation of Ribet’s lemma. An interesting feature of the proof is that it applies to both the residually multiplicity-free and the residually non-multiplicity-free cases. I will discuss an application to a local-global principle for isogenies of elliptic curves.

We explain different kinds of Selmer groups, which are subgroups of Galois cohomology, including Bloch-Kato, strict, and Greenberg Selmer groups. We state part of the Bloch-Kato conjectures and describe a bound joint with A. Betts and M. Leonhardt on the number of rational points on a general higher genus curve, conditional on the Bloch-Kato conjectures. Finally, we explain how to use some Iwasawa theory, specifically Kato’s Euler system and a control theorem of Ochiai, to deduce specific cases of Bloch-Kato associated with elliptic curves.

Lobachevsky started to work on computing volumes of hyperbolic polytopes long before the first model of the hyperbolic space was found. He discovered an extraordinary formula for the volume of an orthoscheme via a special function called dilogarithm.

We will discuss a generalization of the formula of Lobachevsky to higher dimensions. For reasons I do not fully understand, a mild modification of this formula leads to the proof of a conjecture of Goncharov about the depth of multiple polylogarithms. Moreover, the same construction leads to a functional equation for polylogarithms generalizing known equations of Abel, Kummer, and Goncharov.

Guided by these observations, I will define cluster polylogarithms on a cluster variety.

Jakob has kindly agreed to speak about his old work on Arakelov motivic cohomology and comparison with the arithmetic intersection pairing of Gillet-Soul'e.

A universal deformation of Poisson structures was constructed by M.Kontsevich in 90’s. D.Tamarkin explained that the set of universal deformations are in one-to-one correspondence with Drinfeld Associators. On the other hand, we know that all universal deformations of linear Poisson structures are trivial and coincide with universal enveloping algebra. We show that universal deformations of quadratic Poisson structure are as rich as the full set of all deformations.

The first part of the talk will be devoted to the elementary description of Kontsevich Graph complexes and related combinatorics. The relationships with the universal quantization problems of generic and quadratic Poisson structures will be given in the second part of the talk (based on the joint results with Sergei Merkulov https://arxiv.org/abs/2109.07793).

The Arithmetic Fundamental Lemma (AFL) is a local conjecture motivated by decomposing both sides of the Gross—Zagier Formula into local terms using the Relative Trace formula. For each of the local terms, one side is the intersection number in some Rappoport—Zink space. The other side is some orbital integral. To reduce the global computation to local, one needs to consider intersection numbers on both basic and non-basic locus, while the original linear AFL only considers basic locus.

Collaborated with Andreas Mihatsch, we consider the non-basic locus of Unitary Shimura varieties and conjectured a similar version of linear AFL for Rappoport Zink space on non-basic locus parameterizing p-divisible groups with étale extensions. We proved that this version of linear AFL conjecture can be essentially reduced to the linear AFL conjecture for Lubin—Tate spaces, which corresponds to the basic locus parameterizing one-dimensional connected p-divisible groups.

The category of mixed Tate motives over an open integer ring or a number field possesses a notion of p-adic period which diverges somewhat from the complex paradigm: rather than comparing two different fiber functors, it compares two different structures both associated with the same cohomology theory. At first glance, it appears to be a peculiarity of the mixed Tate setting. Yet it plays a central role in the microcosm of mixed Tate Chabauty-Kim. It also connects the study of p-adic iterated integrals with Goncharov’s theory of motivic iterated integrals, and allows us to investigate Goncharov’s conjectures from a p-adic point of view. Lastly, it forms the basis for the so-called p-adic period conjecture. I’ll report on our ongoing work devoted to the construction of p-adic periods beyond the mixed Tate setting, and discuss the possibility of generalizing all aspects of this picture. This is joint work with David Corwin.