A venue for invited speakers to lecture about their work over Zoom. This seminar is organized jointly by Ishai Dan-Cohen (Be’er Sheva) and Ari Shnidman (Jerusalem). The intended format includes 10 minutes for assembly, 60 minutes of lecture, followed by 10 to 20 minutes for questions and informal discussion. To be added to the seminar mailing list, please contact Ishai at ishaida@bgu.ac.il. Zoom coordinates will be sent to the mailing list a few minutes before each meeting.

The seminar meets on Wednesdays, 15:00-16:30, in

This Week


Time: 16:00–17:00

Amnon Besser (Be'er Sheva)

Coleman and Vologodsky integration, height pairings and rational points

Coleman and Vologodsky integration are two related p-adic integration theories. In the first part of the talk I will explain the relation between the theories in some 1-dimensional cases discovered jointly with Sarah Zerbes, and I will conjecture on how this can be generalized, using in part the theory of averages on torsors for unipotent groups developed by Amnon Yekutieli.

In the second part of the talk I will give a new development of the theory of p-adic heights on Abelian varieties, which is in progress with Jan Steffen Muller and Padmavathi Srinivasan. I will explain how the Vologodsky theory gives us a way of computing local contribution to the pairing at each prime and not just at p. Time permitting I will explain the application to rational points, also in the joint work with Steffen and Padma.


2020–21–A meetings

Upcoming Meetings

Date
Title
Speaker
Abstract
Dec 2, 16:00–17:00 Coleman and Vologodsky integration, height pairings and rational pointsOnline Amnon Besser (Be'er Sheva)

Coleman and Vologodsky integration are two related p-adic integration theories. In the first part of the talk I will explain the relation between the theories in some 1-dimensional cases discovered jointly with Sarah Zerbes, and I will conjecture on how this can be generalized, using in part the theory of averages on torsors for unipotent groups developed by Amnon Yekutieli.

In the second part of the talk I will give a new development of the theory of p-adic heights on Abelian varieties, which is in progress with Jan Steffen Muller and Padmavathi Srinivasan. I will explain how the Vologodsky theory gives us a way of computing local contribution to the pairing at each prime and not just at p. Time permitting I will explain the application to rational points, also in the joint work with Steffen and Padma.

Dec 9 TBAOnline Zev Rosengarten (HUJI)
Dec 16 TBAOnline Jakob Scholbach (Münster (Germany))
Dec 23 TBAOnline Mihran Papikian (Pennsylvania State University)
Dec 30 TBAOnline Padmavathi Srinivasan (University of Georgia)
Jan 6 TBAOnline TBA
Jan 13 TBAOnline TBA

Past Meetings

Date
Title
Speaker
Abstract
Oct 21 Recent progress on the Diophantine geometry of curves Minhyong Kim (Warwick)

The study of rational or integral solutions to polynomial equations is among the oldest subjects in mathematics. After a brief description of the history, we will review some recent geometric approaches to describing sets of solutions when the number of variables is 2.

Please click on the link to the “abstract” to view the slides.

Oct 28 Derived categories and birationality Martin Olsson (UC Berkeley )

I will discuss expectations and results around the following question: If $X$ and $Y$ are two smooth projective varieties with equivalent derived categories, when can one conclude that $X$ and $Y$ are birational? The study of Fourier-Mukai equivalences yields many examples of non-birational varieties with equivalent derived categories. On the other hand, it appears that by considering slightly more structure than just the derived categories one can conclude birationality in many cases. This is joint work with Max Lieblich.

Recording available here: https://us02web.zoom.us/rec/share/U7Zp8zsHQrL4WGyhHAx9sSLRNwEPoFAp2AnvK5_lvC4M0_5aByj6YMYM00_zdsiG.Pr97s-6WdDsEx0qM

Nov 4 Quadratic Euler characteristics of hypersurfaces and hypersurface singularities Marc Levine (Essen)

This is a report on joint work with V. Srinivas and Simon Pepin Lehalleur. Recently, with Arpon Raksit, we have shown that for a smooth projective variety X over a field k, the quadratic Euler characteristic of X, an element of the Grothendieck-Witt ring of quadratic forms over k, can be computed via the cup product on Hodge cohomology followed by the canonical trace map. Following work of Carlson-Griffiths, this leads to an explicit formula for the quadratic Euler characteristic of a smooth projective hypersurface defined by a homogeneous polynomial F in terms of the Jacobian ring of F, as well as a similar formula for a smooth hypersurface in a weighted projective space. In some special cases, this leads to quadratic versions of classical conductor formulas with some mysterious and unexpected correction terms, even in characteristic zero.

Nov 11 No meeting No meeting
Nov 18 Rigidity, Residues and Duality: Recent Progress Amnon Yekutieli (Be'er Sheva)

Let K be a regular noetherian ring. I will begin by explaining what is a rigid dualizing complex over an essentially finite type (EFT) K-ring A. This concept was introduced by Van den Bergh in the 1990’s, in the setting of noncommutative algebra. It was imported to commutative algebra by Zhang and myself around 2005, where it was made functorial, and it was also expanded to the arithmetic setting (no base field). The arithmetic setting required the use of DG ring resolutions, and in this aspect there were some major errors in our early treatment. These errors have recently been corrected, in joint work with Ornaghi and Singh. Moreover, we have established the forward functoriality of rigid dualizing complexes w.r.t essentially etale ring homomorphisms, and their backward functoriality w.r.t. finite ring homomorphisms. These results mean that we have a twisted induction pseudofunctor, constructed in a totally algebraic way (rings only, no geometry). Looking to the future, we plan to study a more refined notion: rigid residue complexes. These are complexes of quasi-coherent sheaves in the big etale site of EFT K-rings, and they admit backward functoriality, called ind-rigid traces, w.r.t. arbitrary ring homomorphisms. Rigid residue complexes can be easily glued on EFT K-schemes, and they still have the ind-rigid traces w.r.t. arbitrary scheme maps. The twisted induction now becomes the geometric twisted inverse image pseudofunctor f \mapsto f^!. We expect to prove the Rigid Residue Theorem and the Rigid Duality Theorem for proper maps of EFT K-schemes, thus recovering almost all of the theory in the original book “Residues and Duality”, in a very explicit way. The etale functoriality implies that every finite type Deligne-Mumford (DM) K‑stack admits a rigid residue complex. Here too we have the f \mapsto f^! pseusofunctor. For a map of DM stacks there is the ind-rigid trace. Under a mild technical condition, we expect to prove the Rigid Residue Theorem for proper maps of DM stacks, and the Rigid Duality Theorem for such maps that are also tame. Lecture notes will be available at http://www.math.bgu.ac.il/~amyekut/lectures/RRD-2020/notes.pdf . (November 2020)

Nov 25 Bad reduction and fundamental groups Netan Dogra (Oxford)

This talk will be about two related results concerning Galois actions on pro-p fundamental groups of curves over mixed characteristic local fields, with applications to the algorithmic resolution of Diophantine equations. The first result is joint with Alex Betts, and gives a description of how the Galois action on the fundamental group varies with the choice of basepoint in terms of harmonic analysis on the dual graph of the special fibre of a stable model (when p is different from the residue characteristic). The second result is joint with Jan Vonk, and gives a description of how to compute the Galois action (in a p-adic Hodge theoretic sense) when the residue characteristic is p and the curve has semistable reduction.

Seminar run by Dr. Ishai Dan-Cohen