AGNT Ical Atom

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.

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.