Boston College NT&AG seminar

Abstracts for Spring 2016 ( Older abstracts )

May 5: Ila Varma (Harvard) The average size of 3-torsion elements in ray class groups of quadratic fields
Abstract:
In 1971, Davenport and Heilbronn determined the mean number of 3-torsion elements in the class groups of quadratic fields, when ordered by discriminant. I will describe some aspects of the proof of Davenport and Heilbronn’s theorem; in particular, they prove a relationship via class field theory between the number of 3-torsion ideal classes of quadratic fields and the number of nowhere totally ramified cubic fields over Q. This argument generalizes to give a relationship between 3-torsion elements of the ray class groups of quadratic fields and certain pairs of cubic fields satisfying explicit ramification conditions. I will illustrate how the combination of this fact with Davenport-Heilbronn’s asymptotics on the number of cubic fields of bounded discriminant allows one to compute the mean number of 3-torsion elements in ray class groups of quadratic fields with fixed conductor. If time permits, I will discuss the analogous theorems computing the mean size of 2-torsion elements in ray class groups of cubic fields ordered by discriminant, generalizing Bhargava’s result confirming the only known case of the Cohen-Martinet-Malle heuristics.

Apr 28: Vasily Dolgushev (Temple University) A manifestation of the Grothendieck-Teichmueller group in geometry
Abstract:
Inspired by Grothendieck's lego-game, Vladimir Drinfeld introduced, in 1990, the Grothendieck-Teichmueller group GRT. This group has interesting links to the absolute Galois group of rationals, finite type invariants of tangles, deformation quantization and theory of motives. My talk will be devoted to the manifestation of GRT in the extended moduli of algebraic varieties, which was conjectured by Maxim Kontsevich in 1999. My talk is partially based on the joint paper with Chris Rogers and Thomas Willwacher: http://annals.math.princeton.edu/2015/182-3/p02

Apr 21: Adrian Zahariuc (Harvard) Deformation of Quintic Threefolds to the Chordal Variety
Abstract:
The idea to analyze the geometric or enumerative properties of algebraic curves on Calabi-Yau threefolds by degenerating the underlying threefold has been around for a long time. The first step in applying this method is to understand the space of stable maps to the degenerate fiber. In this talk, I will show an elementary construction of a degeneration of quintic threefolds, for which this step can be carried out in a satisfactory way in the genus zero case. An application to higher genus and some questions relating to Clemens Conjecture will also be discussed.

Apr 07: Alexander Isaev (ANU) Isolated hypersurface singularities and associated forms.
Abstract:
In our recent articles joint with M. Eastwood and J. Alper, it was conjectured that all rational GL_n-invariant functions of homogeneous forms of degree d>2 on complex space {\mathbb C}^n can be extracted, in a canonical way, from those of forms of degree n(d-2) by means of assigning to every form with nonvanishing discriminant the so-called associated form. While this surprising statement is interesting from the point of view of classical invariant theory, its original motivation was the reconstruction problem for isolated hypersurface singularities, which is the problem of finding a constructive proof of the well-known Mather-Yau theorem. Settling the conjecture is part of our program to solve the reconstruction problem for quasihomogeneous isolated hypersurface singularities. In my talk, I will give an overview of the recent progress on the conjecture. If time permits, I will further discuss the morphism that assigns to a nondegenerate form its associated form. This morphism is rather natural and deserves attention regardless of the conjecture. In particular, it leads to a natural contravariant.

Mar 31: Jayce Getz (Duke) Limiting forms of trace formulae and triple product $L$-functions
Abstract:
Langlands has proposed studying limits of trace formulae as an approach to proving Langlands functoriality in general. The proposal has only been carried out in very special cases corresponding more or less to the standard representation and symmetric square representation of GL(2) and the tensor product of GL(2) with itself. We explain how the analytic part of the proposal can be carried out in the case of the tensor product of three copies of GL(2). Time permitting, we discuss the prospects of generalizing the approach to the tensor product of three copies of general linear groups of arbitrary rank.

Mar 17: Scott Mullane (BC) Divisors from the strata of abelian differentials
Abstract:
An abelian differential defines a flat metric with conical singularities at its zeros, such that the underlying Riemann surface can be realised as a polygon whose edges are identified pairwise via translation. A number of questions about geometry and dynamics on Riemann surfaces reduce to studying the strata of abelian differentials with prescribed number and multiplicities of zeros. In this talk we will focus on the divisorial strata that form special codimension-one subvarieties in the moduli space of Riemann surfaces. In particular, we compute the divisor class of such strata closures and relate it to the birational geometry of the Deligne-Mumford compactified moduli space.

Mar 3: Jason Bland (Harvard) Raising the 2-Selmer rank of Jacobians of hyperelliptic curves
Abstract:
A hyperelliptic curve with a rational Weierstrass point has the form y^2 = f(x) with f of degree 2g + 1. I show that the 2-Selmer rank of the quadratic twist dy^2 = f(x) is unbounded as d varies. The arguments are simpler in the case where f is reducible.

Feb. 18: Alexander Perry (Harvard) Derived categories of Gushel-Mukai varieties
Abstract:
I will discuss the derived categories of Fano varieties of Picard number 1, degree 10, and coindex 3. In particular, I will describe an interesting semiorthogonal component of the derived category of such a variety, and discuss its behavior for some special families of fourfolds. This is joint work with Alexander Kuznetsov.

Feb 04: Leo Goldmakher (Williams College) Characters and their nonresidues
Abstract:
Understanding the least quadratic nonresidue (mod p) is a classical problem, with a history stretching back to Gauss. The approach which has led to the strongest results uses character sums, objects which are ubiquitous in analytic number theory. I will discuss character sums, their connection to the least nonresidue, and work of myself and Jonathan Bober (University of Bristol) on a promising new approach to the problem.

Jan 28: Barbara Bolognese (Northeastern) Generic Strange Duality and Verlinde numbers on abelian surfaces
Abstract:
With the purpose of examining some relevant geometric properties of the moduli space of sheaves over an algebraic surface, Le Potier conjectured some unexpected duality between the complete linear series of certain natural divisors, called Theta divisors, on the moduli space. Such conjecture is widely known as Strange Duality conjecture. After having motivated the problem by looking at certain instances of quantization in physics, we will work in the setting of surfaces. We will then sketch the proof in the case of abelian surfaces (obtained in joint work with Marian, Oprea and Yoshioka), giving an idea of the techniques that are used. In particular, we will show how the theory of discrete Heisenberg groups and fiber wise Fourier-Mukai transforms, which might be applied to other cases of interest, enter the picture.

Jan 21: Jarod Alper (ANU) A Luna etale slice theorem for algebraic stacks
Abstract:
Quotient stacks are a distinguished class of algebraic stacks which provide key intuition for studying the geometry of general algebraic stacks. It has long been believed that certain algebraic stacks are in some sense "locally" quotient stacks. In this talk, we will prove that this expectation holds by providing a description of the etale local structure of algebraic stacks near points with linearly reductive stabilizer. We will then discuss a number of striking applications of this result. This is joint work with Jack Hall and David Rydh.

Abstracts for Fall 2015 ( Older abstracts )

Nov. 19: Mihai Fulger (Princeton)Hilbert functions and volumes
Abstract:
When X is a projective variety in P^N, its Hilbert function H(m) = H^0(X; O(m)) is a polynomial of degree dim(X). A geometric generalization of this situation is to take D an arbitrary divisor and to consider the function H_D(m)=H^0(X; O(mD)). In general, this function is not polynomial, but it grows at most like a polynomial of degree dim(X). Its asymptotic growth rate is the volume vol(D). When D is big, i.e. vol(D)>0, and E is effective, then H_{D-E}(m)<=H_D(m)<=H_{D+E}(m) for all m and hence vol(D-E)<=vol(D)<=vol(D+E). It is interesting to study when either equality of volumes holds. In joint work with J. Koll\'ar and B. Lehmann, we show that surprisingly either equality of volumes is equivalent to the equality of Hilbert functions. Koll\'ar has used this to provide a criterion for existence of simultaneous canonical models in families.

Nov. 12: Sam Grushevsky (Stony Brook) Shimura curves contained in the loci of Jacobians of small genus
Abstract:
We construct infinitely many Shimura curves (1-dimensional special subvarieties, which are in particularly totally geodesic) of the moduli space of principally polarized abelian varieties, which are contained in the locus of hyperelliptic Jacobians in genus 3, and in the locus of Jacobians in genus 4. Based on joint work with M. Moeller.

Nov. 05: Beth Romano (BC) The Local Langlands Correspondence: New Examples from the Epipelagic Zone
Abstract:
Let G be a split reductive group over a p-adic field k. The Local Langlands Correspondence (LLC) predicts that for every irreducible supercuspidal representation of G there should be a corresponding field extension of k whose structure reflects certain properties of the representation. The LLC has been proven in many cases for large primes p, but remains mysterious when p is small. In recent work, Jessica Fintzen and I have found new supercuspidal representations for small p, and in some cases I have found the corresponding field extensions. In my talk I will give explicit examples for the case G = G_2.

Oct. 29: Ari Shnidman Selmer groups in families of cubic twists
Abstract:
We present asymptotics for the average size of the 3-Selmer group in cubic twist families of elliptic curves. As a consequence, we conclude that the average size of 3-Selmer is infinite in each such family. We also show that 0% of curves in a given cubic twist family of genus 1 curves are everywhere locally soluble. This is joint work with Manjul Bhargava.

Oct. 22: Fan Gao (Purdue) The Langlands-Shahidi L-functions for Brylinski-Deligne covering groups
Abstract:
Many efforts have been made to extend the Langlands program to central covers of linear reductive groups. For this purpose, a theory of L-groups for covering groups will play a pivotal role as in the linear algebraic case. In the recent foundational work of Martin Weissman, such a theory for Brylinski- Deligne (BD) covering groups is built and the construction is functorial. In this talk, we will give a brief review of the construction. Moreover, for BD covers of arbitrary split reductive group, we show that the constant term of Eisenstein series could be expressed in terms of global (partial) Langlands-Shahidi type L-functions. As in the linear algebraic case, this relies crucially on the analogous of Satake isomorphism, Gindikin-Karpelevich formula and local Langlands correspondence for covering tori. We will give a brief description of these, and meanwhile indicate some immediate questions arising from our result.

Oct. 15: Pitale, Ameya (U. of Oklahoma) Restrictions of modular forms
Abstract:
In this talk, we will consider the problem of restricting Hilbert modular forms to the complex upper half plane. It is easy to see that, on doing the restriction, we get elliptic modular forms. Tonghai Yang conjectured that if we consider a certain set of Hilbert Eisenstein series then the span of their restrictions gives the entire space of elliptic modular forms of a fixed weight and level. We will present progress towards this conjecture. in particular, we will consider the inner product of the restriction of such an Eisenstein series with an elliptic cusp form. This is best done in the adelic setting and leads naturally to local and global Waldspurger models. This is joint work with Rodney Keaton and Yingkun Li.

Oct. 08: Carl Wang Erickson (Brandeis) Modularity of ordinary Galois representations via coarse moduli
Abstract:
Ordinary Galois representations are among those Galois representations expected to arise from modular forms as part of the Langlands philosophy. For Galois representations that are residually irreducible, this follows from various "R=T theorems," where R is a universal ordinary deformation ring for the residual representation and T is an ordinary Hecke algebra. When the residual representation is reducible, the modularity of ordinary Galois representations is still often known, but there is no universal ordinary deformation ring R to correspond to T. This talk will introduce joint work with Preston Wake. We produce a universal deformation ring R for ordinary pseudorepresentations that is a candidate for comparison with T in the residually reducible case. Indeed, we prove that R=T in many cases, and derive some consequences. We will introduce this notion of ordinary pseudorepresentation, and will explain how it can be thought of as the coarse moduli of ordinary representations. This provides evidence toward the idea that coarse moduli rings of Galois representations ought to correspond to Hecke algebras.

Oct. 01: Xiaowei Wang (Rutgers) Moduli space of Fano Kahler-Einstein manifolds
Abstract:
In this talk, we will discuss our construction of compact Hausdorff Moishezon moduli spaces parametrizing smoothable K-stable Fano varieties. The solution relies on the recent solution of the Yau-Tian-Donaldson conjecture by Chen-Donaldson-Sun and Tian. In particular, we prove the uniqueness of the degeneration of Fano Kahler-Einstein manifolds and more algebraic properties that are needed to construct an algebraic moduli space. (This is a joint work with Chi Li and Chenyang Xu).

Sep. 17: Arul Shankar (Harvard) Cohen Lenstra heuristics in thin families of number fields.
Abstract:
e discuss computing the average sizes of the 2-torsion subgroup of class groups in certain thin families of number fields. We show that the averages sometimes differ from the conjectured values. The main tools used are those from the geometry-of-numbers. This are joint works with Manjul Bhargava and Jon Hanke, as well as Wei Ho and Ila Varma.

Sep. 10: Ian Petrow (Ecole Polytechnique) A twisted Motohashi formula
Abstract:
Some of the strongest currently-known subconvex bounds are for L-functions of cusp forms twisted by quadratic characters and are due to Conrey and Iwaniec. Their estimate is derived from an estimate of the cubic moment of those L-functions. I will present a Motohashi-type formula which describes the dual sums of this cubic moment. This Motohashi formula is crucial in extending Conrey and Iwaniec's results to the particularly challenging case of weight 2 cusp forms.

Abstracts for Spring 2015

Apr. 30: Steven J. Miller (Williams College). Finite conductor models for zeros near the central point of elliptic curve L-functions
Abstract:
Random Matrix Theory has successfully modeled the behavior of zeros of elliptic curve L-functions in the limit of large conductors. In this talk we explore the behavior of zeros near the central point for one-parameter families of elliptic curves with rank over Q(T) and small conductors. Zeros of L-functions are conjectured to be simple except possibly at the central point for deep arithmetic reasons; these families provide a fascinating laboratory to explore the effect of multiple zeros on nearby zeros. Though theory suggests the family zeros (those we believe exist due to the Birch and Swinnerton-Dyer Conjecture) should not interact with the remaining zeros, numerical calculations show this is not the case; however, the discrepency is likely due to small conductors, and unlike excess rank is observed to noticeably decrease as we increase the conductors. We shall mix theory and experiment and see some surprisingly results, which leads us to conjecture that an excised orthogonal ensemble correctly models the small conductor behavior

Apr. 23: Chao Li (Harvard) Level raising mod 2 and arbitrary 2-Selmer ranks
Abstract:
We prove a level raising mod p=2 theorem for elliptic curves over Q, generalizing theorems of Ribet and Diamond-Taylor. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families. We will begin by explaining our motivation from W. Zhang's approach to the p-part of the BSD conjecture. Explicit examples will be given to illustrate different phenomena compared to odd p. This is joint work with Bao V. Le Hung.

Apr. 16:Paolo Stellari (Universita degli Studi di Milano) Uniqueness of dg enhancements in geometric contexts
Abstract:
It was a general belief and a formal conjecture by Bondal, Larsen and Lunts that the dg enhancement of the bounded derived category of coherent sheaves or the category of perfect complexes on a (quasi-)projective scheme is unique. This was proved by Lunts and Orlov in a seminal paper. In this talk we will explain how to extend Lunts-Orlov's results to several interesting geometric contexts. Namely, we care about the category of perfect complexes on noetherian separated schemes with enough locally free sheaves and the derived category of quasi-coherent sheaves on any scheme. This is a joint work in progress with A. Canonaco.

Apr. 9:Yuchen Zhang (Michigan) The volume of isolated singularities
Abstract:
Boucksom, de Fernex and Favre defined a non-log-canonical volume to study isolated singularities and developed several geometric results. Their definition is based on the intersection number of nef envelope of the log discrepancy b-divisor. In this talk, I will give an alternate definition of this volume by log canonical modification.

Mar. 26:Sho Tanimoto (Rice) Height zeta functions of algebraic groups
Abstract:
One of fundamental questions in diophantine geometry is to understand asymptotic formulas for counting functions of rational points of bounded height, and for each counting function, one can associate a Dirichlet series socalled height zeta functions. Height zeta functions have been studied for a variety of equivariant compactifications of homogeneous spaces including toric varieties and equivariant compactifications of vector groups. In this talk, I will review some results in this area, and report a recent progress toward understanding of height zeta functions of one-sided equivariant compactifications of algebraic groups. This is joint work with Ramin Takloo-Bighash

Mar. 12:Michael Magee (IAS) Expanding maps and continued fractions
Abstract:
In this talk I'll discuss a lattice point count for a thin semigroup inside SL_2(Z). It is important for applications that I'll describe that one can perform this count uniformly throughout congruence classes. The approach to counting is dynamical - with input from both the real place and finite primes. At the real place methods of Dolgopyat that were further built upon by Naud are at play. At finite primes the necessary mixing property follows from work of Bourgain and Gamburd that uses tripling estimates in SL_2(F_p) due to Helfgott as key input. This is joint work with Hee Oh and Dale Winter.

Feb. 19: Charlotte Chan (U. of Michigan), ) p-adic Deligne-Lusztig constructions and the local Langlands correspondence
Abstract:
The representation theory of SL2(Fq) can be studied by studying the geometry of the Drinfeld curve. This is a special case of Deligne-Lusztig theory, which gives a beautiful geometric construction of the irreducible representations of finite reductive groups. I will discuss recent progress in studying Lusztig's conjectural construction of a p-adic analogue of this story. It turns out that for division algebras, the cohomology of the p-adic Deligne-Lusztig (ind-)scheme gives rise to supercuspidal representations of arbitrary depth and furthermore gives a geometric realization of the local Langlands and Jacquet-Langlands correspondences. (This talk is based on arXiv:1406.6122 and forthcoming work.)

Feb. 5: Tony Varilly-Alvarado (Rice) Special cubic fourfolds and K3 surfaces: an arithmetic perspective
Abstract:
Cubic fourfolds containing a surface not homologous to a complete intersection often have nonspecial cohomology isomorphic to the primitive cohomology of a K3 surface "twisted" by an element of the Brauer group. This isomorphism is usually a manifestation of a geometric correspondence, which has consequences for the distribution of rational points on K3 surfaces over number fields. We will discuss this circle of ideas, including some recent developments in joint work with McKinnie, Sawon and Tanimoto on p-torsion Brauer classes of K3 surfaces and with Tanimoto on the Kodaira dimension of the moduli space of special cubic fourfolds of fixed discriminant

Jan. 29: Alex Kontorovich (Rutgers) Equidistribution of Shears and Applications
Abstract:
A "shear" is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete group (possibly of infinite co-volume). In joint work with Dubi Kelmer, we prove the regularized equidistribution of shears under large translates. We give applications including to moments of GL(2) automorphic L-functions, and to counting integer points on affine homogeneous varieties. No prior knowledge of these topics will be assumed.

Jan. 15: Ethan Cotterill (Universidade Federal Fluminense) Dimension counts for singular rational curves
Abstract:
Rational curves are essential tools for classifying algebraic varieties. Establishing dimension bounds for families of embedded rational curves that admit singularities of a particular type arises arises naturally as part of this classification. Singularities, in turn, are classified by their value semigroups. Unibranch singularities are associated to numerical semigroups, i.e. sub-semigroups of the natural numbers. These fit naturally into a tree, and each is associated with a particular weight, from which a bound on the dimension of the corresponding stratum in the Grassmannian may be derived. Understanding how weights grow as a function of (arithmetic) genus g, i.e. within the tree, is thus fundamental. We establish that for genus g \leq 8, the dimension of unibranch singularities is as one would naively expect, but that expectations fail as soon as g=9. Multibranch singularities are far more complicated; in this case, we give a general classification strategy and again show, using semigroups, that dimension grows as expected relative to g when g \leq 5. This is joint work with Lia Fusaro Abrantes and Renato Vidal Martins.