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

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.

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.

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.

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.