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 14:10 sharp and end at 15:10. Meetings are held in the subterranean room -101. We expect to broadcast most meetings over Zoom at the URL

https://us02web.zoom.us/j/84776059631?pwd=D1BRE1heMcDazcQCZ70Aa6auiaI6ip.1

Meeting ID: 847 7605 9631 Passcode: 521044

However, if at all possible, attendees are asked to come in person.

The seminar meets on Wednesdays, 14:10-15:10, in -101

This Week


Time: 15:10–16:10

Borys Kadets (HUJI)

Groups of points on abelian and Jacobian varieties over finite fields. Please note the unusual time!

I will describe various results, some old and some new, on the structure of the groups of points of an abelian variety over a finite field. The talk will focus on the case of varieties of large dimension over a fixed finite field. In this regime, the Weil bounds allow for the possibility of the exponent of the group staying bounded as the dimension grows. I will explain that at least in the case of Jacobians this cannot be the case. Part of the talk is based on recent joint work with Daniel Keliher.


2024–25–A meetings

Upcoming Meetings

Date
Title
Speaker
Abstract
Nov 27 TBAOnline No Meeting
Dec 4 tbdOnline Yotam Hendel (BGU)

tbd

Dec 11 TBAOnline
Dec 18 TBAOnline Gal Binyamini (tbc) (Weizmann)
Dec 25 TBAOnline Shay Ben Moshe (Weizmann)
Jan 1 TBAOnline
Jan 8 TBAOnline Amnon Yekutieli (BGU)
Jan 15 TBAOnline
Jan 22 TBAOnline
Jan 29 TBAOnline

Past Meetings

Date
Title
Speaker
Abstract
Nov 6 TQFTs for pro-p Poincare duality groups Nadav Gropper (University of Haifa)

In the talk, I will discuss the Turner-Turaev formalism for unoriented Topological Quantum Field Theory (TQFT). Building upon this formalism, I will introduce an analogous version for (d+1)-dimensional TQFT for pro-p Poincare duality groups. In the case of d = 1, this enables us to study cobordisms and TQFTs for both the maximal pro-p quotient of absolute Galois groups of p-adic fields and pi_1(X)^p, the pro-p completions of fundamental groups of surfaces. This generalisation gives a framework for arithmetic TQFTs and strengthens the analogies within arithmetic topology, which relates p-adic fields to surfaces (oriented mod p^r). I will explain the classification of TQFTS for the (1+1)-dimensional case, in terms of Frobenius algebras with some extra structure.

If time permits, I will explain how we define a Dijkgraaf Witten like theory, to get formulas for counting G-covers of X, where X is either a surface, or a p adic field, and G is a p-group (these formulas are similar to the ones given by Mednykh for surfaces using TQFTs, and by Masakazu Yamagishi using a more algebraic approach). I will also try to outline how we plan to also get similar formulas for Hom(\pi_1(X)^p,G), where G=GL_n(k) for k=F_{p^r} or Z/p^rZ.

The talk is based on joint work with Oren Ben-Bassat.

Nov 13 The power operation in the Galois cohomology of a reductive group over a number field Mikhail Borovoi (TAU)

For a number field $K$ admitting an embedding into the field of real numbers $\mathbb{R}$, it is impossible to construct a functorial in $G$ group structure in the Galois cohomology pointed set $H^1(K,G)$ for all connected reductive $K$-groups $G$. However, over an arbitrary number field $K$, we define a diamond (or power) operation of raising to power $n$

$$ (x,n) \mapsto x^{\Diamond n}: H^1(K,G) \times Z \to H^1(K,G). $$

We show that this operation has many functorial properties. When $G$ is a torus, the set $H^1(K,G)$ has a natural group structure, and $x^{\Diamond n}$ coincides with the $n$-th power of $x$ in this group.

For a cohomology class $x \in H^1(K,G)$, we define the period $\operatorname{per}(x)$ to be the greatest common divisor of $n>0$ such that $x^{\Diamond n}=1$, and the index $\operatorname{ind}(x)$ to be the greatest common divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting $x$. These period and index generalize the period and index of a central simple algebra over $K$ (in the special case where $G$ is the projective linear group $\operatorname{PGL}_n$, the elements of $H^1(K, G)$ can be represented by central simple algebras). For an arbitrary reductive group $G$ defined over a local or global field $K$, we show that $\operatorname{per}(x)$ divides $\operatorname{ind}(x)$, that $\operatorname{per}(x)$ and $\operatorname{ind}(x)$ have the same prime factors, but the equality $\operatorname{per}(x) = \operatorname{ind}(x)$ may not hold.

The talk is based on joint work with Zinovy Reichstein. All necessary definitions will be given, including the definition of the Galois cohomology set $H^1(K,G)$.

Nov 20, 15:10–16:10 Groups of points on abelian and Jacobian varieties over finite fields. Please note the unusual time! Borys Kadets (HUJI)

I will describe various results, some old and some new, on the structure of the groups of points of an abelian variety over a finite field. The talk will focus on the case of varieties of large dimension over a fixed finite field. In this regime, the Weil bounds allow for the possibility of the exponent of the group staying bounded as the dimension grows. I will explain that at least in the case of Jacobians this cannot be the case. Part of the talk is based on recent joint work with Daniel Keliher.