Algebra-Number Theory Archives for Academic Year 2017

Shimura varieties for unitary groups and a question of sign

When: Wed, September 13, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Michael Rapoport (UMCP) -
Abstract: Shimura varieties associated to groups of unitary similitudes are well-known to be closely related to moduli spaces of abelian varieties with additional structure. However, only under very favorable circumstances do these moduli schemes yield arithmetic models of these Shimura varieties, as was proved by Kottwitz. I will show that by a small central modification of the unitary group, one obtains Shimura varieties with better properties.

Linked Grassmannians and Local Models of Shimura Varieties

When: Mon, October 9, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Brian Hwang (Cornell)
Abstract: We will show how linked Grassmannians, a class of moduli spaces for
limit linear series on algebraic curves, are related to local models
of Shimura varieties. For certain questions, linked Grassmannians
provide an easy formalism for moduli-theoretic aspects of the theory
of local models, in terms of degenerations of flag varieties. In
particular, we'll show that local models for certain classes of
Shimura varieties of PEL-type are linked Grassmannians in disguise,
introduce a dictionary between the two types of objects, and point to
some questions that this raises. For example, both linked
Grassmannians and local models come with a number of natural
stratifications, but are they related to each other? This is joint
work with Binglin Li.

Algebraic Combinatorics in Geometric Complexity Theory

When: Wed, November 1, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Greta Panova (Institute for Advanced Study) -
Abstract: Some of the oldest classical problems in Algebraic Combinatorics concern finding a "combinatorial interpretation", or, more formally, a #P formula, of structure constants and multiplicities arising naturally in Representation Theory and Algebraic Geometry. Among them is the 80-year old problem of Murnaghan to find a positive combinatorial formula for the Kronecker coefficients of the Symmetric Group. More recently this and other related problems emerged in the newer area of Geometric Complexity Theory -- a program of Mulmuley--Sohoni aimed at resolving computational complexity problems like the P vs NP problem (or more precisely its algebraic version VP vs VNP) via Representation Theory and Algebraic Geometry.

We will describe what this is all about from Combinatorics to Complexity, and show how the little we know about Kronecker coefficients can still be used to show that the P vs NP problem is even harder to solve than originally expected.

[This talk will feature results from joint works with Peter B\"urgisser, Fulvio Gesmundo, Christian Ikenmeyer, Igor Pak.]

Shifted Sums of the Theta Function

When: Mon, November 6, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Thomas Hulse (Morgan State University) -
Abstract: We consider the theta function, more specifically the half-integral weight holomorphic form studied by Shimura. The Fourier coefficients of this object, and powers of it, encode information about when an integer can be written as the sum of squares and so can be used to investigate classical problems from number theory. In this talk, we will discuss shifted sums of these Fourier coefficients, derived as variants of the Rankin-Selberg convolution, and show how these can be used to study generalizations and variants of the Gauss Circle Problem and may provide a new avenue for investigating the Congruent Number Problem. This is joint work with Chan Ieong Kuan, David Lowry-Duda and Alexander Walker.

On the arithmetic Todd class of Hilbert modular surfaces and the Jacquet-Langlands correspondence

When: Mon, November 13, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Gerard Freixas i Monplet (Institut de Mathematiques, Jussieu, Paris) -
Abstract: I will report on joint work with Siddarth Sankaran, on some compatibility of the Riemann-Roch formula in Arakelov geometry and the Jacquet-Langlands correspondence. More precisely, we consider a twisted Hilbert modular surface and its arithmetic Todd class (an arithmetic version of the usual volume of the surface). The Riemann-Roch formula relates this to automorphic data, both of holomorphic and non-holomorphic nature. It is tempting to experiment with the Jacquet-Langlands correspondence to, in its turn, relate the latter to similar data on a Shimura curve over a real quadratic field. And then apply Riemann-Roch again to pass to the arithmetic Todd class of the Shimura curve. This should lead to an identity of arithmetic Todd classes, that one can actually check by explicit computation. This explains a formal coincidence of a result of Yuan-Zhang for the Shimura curve, and of ours for the twisted Hilbert modular surface. It also indicates a relation between the Riemann-Roch theorem and the trace formula, that we don’t understand. This work was also motivated by similar considerations with the classical Hilbert modular surfaces, for which the Riemann-Roch formula is not known but can be conjectured (this is ongoing joint work with Deniis Eriksson and Siddarth Sankaran).

Degeneracy loci and nested Hilbert schemes

When: Mon, November 20, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Amin Gholampour (University of Maryland) -
Abstract: We discuss situations in which the degeneracy locus of a map of vector bundles carries a natural perfect obstruction theory whose virtual cycle can be calculated by the Thom-Porteous formula. This generalizes the well-known case of the zero locus of a section of a vector bundle.
We apply this to nested Hilbert schemes of points and curves on surfaces. The resulting virtual cycles agree with the ones coming from Vafa-Witten theory and the reduced localized local Donaldson-Thomas theory of surfaces. This enables us to express some of these invariants in terms of Carlsson-Okounkov operators.

A mod-p Artin-Tate conjecture and generalized Herbrand-Ribet.

When: Wed, November 29, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Dipendra Prasad (UMD and TIFR (India)) -

Abstract: Following the natural instinct that when a group operates on a number
field then every term in the class number formula should factorize ‘compatibly’ ac-
cording to the representation theory (both complex and modular) of the group, we are
led — in the spirit of Herbrand-Ribet’s theorem on the p-component of the class num-
ber of Q(\zeta_p) — to some natural questions about the p-part of the classgroup of any CM
Galois extension of Q as a module for Gal(K/Q), and about integrality of L-values.
This talk will attempt doing this in terms of precise conjectures. Talk is based on a recent paper
with the same title available in the arXiv.

A New Northcott Property for Faltings Height

When: Mon, December 4, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Lucia Mocz (Princeton University) -
Abstract: The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.

Cycles in the de Rham cohomology of abelian varieties over number fields

When: Mon, December 18, 2017 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Yunqing Tang (Princeton University and IAS, Princeton) -

Abstract: In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus and Blasius. Ogus predicted that all such cycles are Hodge. In this talk, I will first introduce Ogus' conjecture as a crystalline analogue of Mumford--Tate conjecture and explain how a theorem of Bost on algebraic foliation is related. After this, I will discuss the proof of Ogus' conjecture for some families of abelian varieties.

Ramification Theory for Arbitrary Valuation Rings in Positive Characteristic.

When: Mon, February 19, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Vaidehee Thatte (Queen's University, Ontario, Canada) -
Abstract: In classical ramification theory, we consider extensions of complete discrete
valuation rings with perfect residue fields. We would like to study
arbitrary valuation rings with possibly imperfect residue fields and possibly
non-discrete valuations of rank >=1, since many interesting complications
arise for such rings. In particular, defect may occur (i.e. we can have a
non-trivial extension, such that there is no extension of the residue field or
the value group).
We present some new results for Artin-Schreier extensions of arbitrary
valuation fields in positive characteristic p. These results relate the \higher
ramification ideal" of the extension with the ideal generated by the inverses
of Artin-Schreier generators via the norm map. We also introduce a general-
ization and further refinement of Kato's refined Swan conductor in this case.
Similar results are true in mixed characteristic (0; p).

The arithmetic of arithmetic Coxeter groups

When: Wed, February 28, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Marty Weissman (UCSC) -

Abstract: In the 1990s, John H. Conway developed a visual approach to the study of integer-valued binary quadratic forms. His creation, the "topograph," sheds light on classical reduction theory, the solution of Pell-type equations, and allows tedious algebraic estimates to be simplified with straightforward geometric arguments. The geometry of the topograph arises from a coincidence between the Coxeter group of type (3, infinity) and the group PGL(2,Z). From this perspective, Conway's topograph is the first in a series of applications arising from coincidences between Coxeter groups and arithmetic groups. In this talk, I will survey Conway's results and generalizations arising from arithmetic hyperbolic Coxeter groups.

Pairings arising from a cup product in the Galois cohomology of number fields (candidacy exam)

When: Mon, March 26, 2018 - 2:00pm
Where: MTH 1310

Speaker: Steven Reich (UMd) -

Chow motives associated to certain algebraic Hecke characters

When: Mon, April 2, 2018 - 2:00pm
Where: Kirwan 3206
Speaker: Laure Flapan (Northeastern) -
Abstract: The Langlands philosophy predicts a correspondence between certain automorphic representations and Galois representations, while the philosophy surrounding the Fontaine-Mazur conjecture specifies when such Galois representations arise from the l-adic cohomology of a variety or, more generally, a motive. Given that one can attach L-functions to both motives and automorphic forms, one would like to understand how to associate a motive to a given automorphic form in order to yield an equality of their respective L-functions. In this talk, we explore this question in the case of powers of certain algebraic Hecke characters and certain CM motives. This is joint work with Jaclyn Lang.

Regarding the Connectedness of Degeneracy Loci: Reflections on a Paper of Fulton and Lazarsfeld (candidacy exam)

When: Wed, April 4, 2018 - 2:00pm
Where: MTH 1313

Speaker: David Pincus (UMd) -
Abstract: Discussion of a paper in Acta Math. 146, 271-283 (1981)

Lang-Trotter Questions on the Reductions of Abelian Varieties (thesis defense)

When: Mon, April 9, 2018 - 2:00pm
Where: MTH 1310
Speaker: Samuel Bloom (UMCP)

Asymptotically log del Pezzo surfaces

When: Thu, April 19, 2018 - 12:00pm
Where: Kirwan Hall 0401
Speaker: Jesus Martinez-Garcia (University of Bath) -
Abstract: Asymptotically log del Pezzo surfaces are a generalisation of del Pezzo surfaces. These surfaces of Fano type are rational and belong to an infinite number of deformation families. Moreover, they are natural candidates for the existence of singular Kaehler-Einstein metrics with conical singularities of large angles along a divisor. Asymptotically log del Pezzo surfaces were introduced by Cheltsov and Rubinstein who classified them under some extra strong condition. We remove the need for an extra condition to give a full classification of these surfaces. This is joint work with Yanir Rubinstein.

A numerical invariant for representations of algebras

When: Mon, April 23, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Zinovy Reichstein (University of British Columbia) -
Abstract: A classical theorem of Brauer asserts that every finite-dimensional
non-modular representation ρ of a finite group G defined over a
field K, whose character takes values in a subfield k, descends to k,
provided that k has suitable roots of unity. If k does not contain
these roots of unity, it is natural to ask how far ρ is from being
definable over k. The classical answer is given by the Schur index of
ρ, which is the smallest degree of a finite field extension l/k
such that ρ can be defined over l. In this talk, based on joint
work with Nikita Karpenko, Julia Pevtsova and Dave Benson, I will
discuss another invariant, the essential dimension of ρ. This
invariant measures "how far" ρ is from being definable over k in a
different way, by using transcendental, rather than algebraic field
extensions. I will also discuss related work on representations of
algebras, due to Federico Scavia.

A gentle approach to the de Rham-Witt complex

When: Mon, April 30, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Akhil Mathew (University of Chicago) -
Abstract: The de Rham-Witt complex of a smooth algebra over a perfect
field provides a chain complex representative of its crystalline
cohomology, a canonical characteristic zero lift of its algebraic de
Rham cohomology. We describe a simple approach to the construction of
the de Rham-Witt complex. This relates to a homological operation
L\eta_p on the derived category, introduced by Berthelot and Ogus, and
can be viewed as a toy analog of a cyclotomic structure. This is joint
work with Bhargav Bhatt and Jacob Lurie.

Two problems connected to the Riemann Hypothesis

When: Tue, May 1, 2018 - 11:00am
Where: Kirwan Hall 3206
Speaker: Don Zagier (Max Planck Institute for Mathematics) -
Abstract: Just to clarify, I have not proved RH, but without having ever planned
to do so found myself a coauthor on two completely different
papers, both rather fun, where the RH or GRH plays the key role,
one with John Lewis on a strange equivalence to GRH involving
cotangent sums and "quantum modular forms" and one with Ken Ono
and two of his postdocs on a proof of a weak version of an old
conjecture of Polya that in its strong version would imply RH.

Goldfeld's conjecture and congruences between Heegner points

When: Mon, May 7, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Chao Li (Columbia University) -
Abstract: Given an elliptic curve E over Q, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever E has a rational 3-isogeny. We also prove the analogous result for the sextic twists of j-invariant 0 curves. For a more general elliptic curve E, we show that the number of quadratic twists of E up to twisting discriminant X of analytic rank 0 (resp. 1) is >> X/log^{5/6}X, improving the current best general bound towards Goldfeld's conjecture due to Ono--Skinner (resp. Perelli--Pomykala). We prove these results by establishing a congruence formula between p-adic logarithms of Heegner points based on Coleman's integration. This is joint work with Daniel Kriz.

Connectedness of Degeneracy Loci (candidacy exam, part 2)

When: Wed, May 23, 2018 - 2:00pm
Where: Kirwan Hall 1311

Speaker: David Pincus (UMd) -
Abstract: Discussion of a paper in Acta Math. 146, 271-283 (1981)