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 Mondays, 12:10-13:10, in -101

2022–23–B meetings

Mar 27 An Algebraic Approach to the Cotangent Complex (online meeting) Amnon Yekutieli (BGU)

Let $B/A$ be a pair of commutative rings. We propose an algebraic approach to the cotangent complex $L_{B/A}$. Using commutative semi-free DG ring resolutions of B relative to A, we construct a complex of $B$-modules $LCot_{B/A}$. This construction works more generally for a pair $B/A$ of commutative DG rings.

In the talk we will explain all these concepts. Then we will discuss the important properties of the DG $B$-module $LCot_{B/A}$. It time permits, we’ll outline some of the proofs.

It is conjectured that for a pair of rings $B/A$, our $LCot_{B/A}$ coincides with the usual cotangent complex $L_{B/A}$, which is constructed by simplicial methods. We shall also relate $LCot_{B/A}$ to modern homotopical versions of the cotangent complex.

Slides: https://sites.google.com/view/amyekut-math/home/lectures/cotangent

May 29 Some cases of the Zilber-Pink conjecture for curves in $\mathcal{A}_g$ George Papas (HUJI)

The Zilber-Pink conjecture is a far reaching and widely open conjecture in the field of unlikely intersections generalizing many previous results in the area such as the Andre-Oort conjecture. We discuss this conjecture and how some cases of it can be established for curves in $\mathcal{A}_g$, the moduli space of principally polarized g-dimensional abelian varieties, following the Pila-Zannier strategy and bounds for the values of the Weil height at certain exceptional points of the curve.

Jun 5 l-Adic local systems and Higgs bundles Hongjie Yu (Weizmann)

In 1981, Drinfeld enumerated the number of irreducible l-adic local systems of rank two on a projective smooth curve in positive characteristic fixed by the Frobenius endomorphism. Interestingly, this number bears resemblance to the number of points on a variety over a finite field. Deligne proposed conjectures to extend and comprehend Drinfeld’s result. In this talk, I will present Deligne’s conjectures and discuss some mysterious phenomena that have emerged in various cases where this number is related to the number of stable Higgs bundles.

Jun 19 TBA David Ter-Borch Gram Lilienfeldt (HUJI)

The Gross-Zagier formula equates (up to an explicit non-zero constant) the central value of the first derivative of the Rankin-Selberg L-function of a weight 2 eigenform and the theta series of a class group character of an imaginary quadratic field (satisfying the Heegner hypothesis) with the height of a Heegner point on the corresponding modular curve. This equality is a key ingredient in the proof of the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rationals in analytic rank 0 and 1. Two important generalizations present themselves: to allow eigenforms of higher weight, and to allow Hecke characters of infinite order. The former one is due to Shou-Wu Zhang. The latter one is the subject of a joint work in progress with Ari Shnidman and requires the calculation of the Beilinson-Bloch heights of generalized Heegner cycles. In this talk, I will report on the calculation of the archimedean local heights of these cycles.

Seminar run by Dr. Ishai Dan-Cohen