Welcome to the Maryland Distinguished Lectures in Algebra and Number Theory! This is an annual lecture series concerning recent developments in Algebra and Number Theory, delivered by the world's foremost experts. It is organized by Thomas Haines.
2022 Mark Kisin
Theme: Arithmetic of abelian varieties and their moduli
Essential dimension and prismatic cohomology (Colloquium)
Wednesday, December 7 at 3:15pm
University of Maryland - Kirwan Hall Rm 3206
Abstract: The smallest number of parameters needed to define an algebraic covering space is called its essential dimension. Questions about this invariant go back to Klein, Kronecker and Hilbert and are related to Hilbert's 13th problem. In this talk, I will give a little history, and then explain a new approach which relies on recent developments in $p$-adic Hodge theory. This is joint work with Benson Farb and Jesse Wolfson.
Frobenius conjugacy classes attached to abelian varieties
Thursday, December 8 at 2:00pm
University of Maryland - Kirwan Hall Rm 3206
Abstract: The Mumford-Tate group of an abelian variety $A$ over the complex numbers is an algebraic group $G$, defined in terms of the complex geometry of $A$, more specifically its Hodge structure. If $A$ is defined over a number field $K$, then a remarkable result of Deligne asserts that the ℓ -adic cohomology of $A$ gives rise to a $G$ -valued Galois representation ρℓ:Gal($\bar K/K$)→$G(\mathbb Q_\ell)$. We will show that for a place of good reduction v∤ℓ of A, the conjugacy class of Frobenius ρℓ(Frobv) does not depend on ℓ. This is joint work with Rong Zhou.
Heights in the isogeny class of an abelian variety
Friday, December 9 at 2:00pm
University of Maryland - Kirwan Hall Rm 3206
Abstract: Let $A$ be an abelian variety over $\bar{\mathbb Q}.$ In this talk I will consider the following conjecture of Mocz. Conjecture: Let $c > 0.$ In the isogeny class of $A,$ there are only finitely many isomorphism classes of abelian varieties of height $ c.$ I will sketch a proof of the conjecture when the Mumford-Tate conjecture - which is known in many cases - holds for $A.$ This result should be compared with Faltings' famous theorem, which is about finiteness for abelian varieties defined over a fixed number field. This is joint work with Lucia Mocz.
2021 Wei Zhang
December 1, 2021 - Cycles on products of elliptic curves and a conjecture of Bloch-Kato
Wednesday, December 1 at 3:15pm
University of Maryland - Kirwan Hall Rm 3206
Abstract: One classical way to study rational or integral solutions of a polynomial equation is to look at the simpler questions of the various congruence equations modulo n for all integer n. The conjecture of Birch and Swinnerton-Dyer predicts that, for elliptic curves (defined by equations of the form y^2=x^3+ax+b with integer coefficients), the data from these congruence equations mod n should actually encode much information on the solutions in rational numbers. In the first talk we will discuss a generalization of the question to the product of several elliptic curves, where, instead of rational points, we look for algebraic cycles (i.e., parameter solutions) modulo suitable equivalence relations (rational equivalence, Abel—Jacobi or its p-adic version). In particular, I'll report some recent results on a conjecture of Bloch-Kato.
December 2, 2021 - Periods of automorphic forms and L-values
Thursday, December 2 at 11:00am
University of Maryland - Kirwan Hall Rm 3206
Abstract: In the second and the third talk I will report some of the ingredients in the proof of the results presented in the first talk. For a reductive group G over a number field and H a suitable subgroup, a fundamental question is to study the period integral of an automorphic form on G along the subgroup H. I’ll give examples with an emphasis on the cases that are related to special values of L-functions.
December 3, 2021 - Heights of special cycles on Shimura varieties and L-derivatives
Friday, December 3 at 2:00pm
University of Maryland - Kirwan Hall Rm 3206
Abstract: In the last talk, I will move to an arithmetic analog of period integrals from the second talk. Here we consider Shimura varieties defined by a group G and special algebraic cycles defined by a subgroup H of G. The height pairing of such special cycles is often related to the special value of the first derivative of L-function; the first of such instance was proved by Gross and Zagier in 1980s. We report the progress on the arithmetic Gan-Gross-Prasad conjecture, one of the generalizations of Gross—Zagier theorem to high dimensional varieties.
2019 Ngô Bảo Châu
December 4, 2019 - On The Hitchin Fibration
On The Hitchin Fibration
Wednesday, December 4 at 3:15pm
Chicago
Abstract: Simpson's non-Abelian Hodge theory stipulates a diffeomorphism between the moduli space of flat connections on a smooth projective variety and the moduli space of semi-stable Higgs bundles with trivial Chern classes. The main feature of the moduli space of Higgs bundles is the Hitchin map calculating the characteristic polynomial of the Higgs field. Over a curve, the structure of the Hitchin map is fairly well understood as an abelian fibration with degeneration. When the base field is a finite field, counting points on the Hitchin fibration allows us to connect the geometry of the Hitchin fibration with orbital integrals and the trace formula. This interplay between geometry and harmonic analysis has been very fruitful for understanding both sides of the story, and in particular, it gave rise to a proof for the fundamental lemma. I will give an account of this interplay in my first lecture.
December 5, 2019 - On The Hitchin Fibration II
On The Hitchin Fibration II
Thursday, December 5 at 2:00 pm
Chicago
Abstract: In the second lecture, I want to discuss the theory of non-archimedean integration on the Hitchin fibration due to Groechenig, Wyss and Ziegler. Surprisingly, calculating nonarchimedean integrals is not exactly the same as counting points and this approach gives another proof of the fundamental lemma, and this discrepancy sheds yet new lights on the theory of endoscopy. The proof is also more elementary in the sense that it does not use the theory of perverse sheaves.
December 6, 2019 - On The Hitchin Fibration III
On The Hitchin Fibration III
Friday, December 6 at 2:00 pm
Chicago
Abstract: In my third lecture, I want to report on a completely different development on the moduli space of Higgs bundles. In joint work with T.H. Chen we started exploring the structure of the Hitchin map for the moduli space of Higgs bundles over higher-dimensional varieties, which raises interesting questions on the geometry of commuting varieties.
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