A venue for invited and local speakers to present their research on topics surrounding algebraic geometry and number theory, broadly conceived. All meetings start at 16:00 sharp and end at 17:00. Some meetings are held in the subterranean room -101, others are held online. We expect to broadcast most meetings over Zoom at the URL https://us02web.zoom.us/j/84396320442

The seminar meets on Wednesdays, 16:00-17:00, in -101

This Week


David Corwin (BGU)

Bloch-Kato Groups and Iwasawa Theory in Chabauty-Kim

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.


2021–22–B meetings

Upcoming Meetings

Date
Title
Speaker
Abstract
May 25 Bloch-Kato Groups and Iwasawa Theory in Chabauty-KimOnline David Corwin (BGU)

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.

Jun 1 TBAOnline Daniil Rudenko (online meeting) (Chicago)
Jun 8 TBAOnline Daniil Rudenko (online meeting) (Chicago)
Jun 15 Arakelov motivic cohomologyOnline Jakob Scholbach (online meeting) (Munster)

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.

Jun 22 TBAOnline Qirui Li (online meeting) (Bonn)
Jun 29 TBAOnline Ishai Dan-Cohen (BGU)

Past Meetings

Date
Title
Speaker
Abstract
Mar 30 Isogenous (non-)hyperelliptic CM Jacobians: constructions, results, and Shimura class groups. (-101) Bogdan Adrian Dina (HUJI)

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.

Apr 6 TBA No talk
Apr 13 TBA No meeting
Apr 20 TBA No meeting
Apr 27 A Constructive Approach to Derived Algebra, online meetingOnline Amnon Yekutieli (BGU)

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)

May 4 TBA No meeting
May 11 Introduction to Diophantine approximation and a generalisation of Roth’s theorem Paolo Dolce (BGU)

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.

May 18 Ribet’s lemma for GL_2 modulo prime powers Amit Ophir (HUJI)

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.

Seminar run by Dr. Ishai Dan-Cohen