NTS ABSTRACTFall2018

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Sept 6

Simon Marshall

What I did in my holidays

Abstract: I will discuss progress I made on the subconvexity problem for automorphic L-functions while at the IAS this past year.

Sept 13

Nigel Boston

2-class towers of cyclic cubic fields

Abstract: The Galois group of the p-class tower of a number field K is a somewhat mysterious group. With Bush and Hajir, I introduced heuristics for how often this group is isomorphic to a given finite p-group G (p odd) as K runs through all imaginary (resp. real) quadratic fields. The next case of interest is to let K run through all cyclic cubic fields. The case p=2 already introduces new phenomena as regards the distribution of p-class groups (or G^{ab}), because of the presence of pth roots of 1 in K. Our investigations indicate further new phenomena when investigating the distribution of p-class tower groups G in this case. Joint work with Michael Bush.

Sept 20

Naser T. Sardari

Bounds on the multiplicity of the Hecke eigenvalues

Abstract: Fix an integer N and a prime p\nmid N where p> 3. Given any p-adic valuation v_p on \bar{\mathbb{Q}} (normalized with v_p(p)=1) and an algebraic integer \lambda \in \bar{\mathbb{Q}}; e.g., \lambda=0, we show that the number of newforms f of level N and even weight k such that T_p(f)=\lambda f is bounded independently of k and only depends on v_p(\lambda) and N.

Sept 27

Florian Ian Sprung

How does the Mordell-Weil rank of an elliptic curve grow in towers of number fields, if you start with a quadratic imaginary field?

Abstract: The Mordell-Weil theorem says that for a number field $K$, the $K$-rational solutions of an elliptic curve $E: y^2=x^3+ax+b$ form a finitely generated abelian group. One natural question is how the rank of these groups behave as $K$ varies. In infinite towers of number fields, one may guess that the rank should keep growing ('more numbers should mean new solutions'). However, this guess is not correct: When starting with $\Q$ and climbing along the tower of number fields that make up the $Z_p$-extension of $Q$, the rank stops growing, as envisioned by Mazur in the 1970's. What happens if we start with an imaginary quadratic field instead of $\Q$, and look at the growth of the rank in a $Z_p$-extension? Our new found 'intuition' now tells us we should expect the rank to stay bounded, but this is not always the case, as shown in Bertolini's thesis. So how badly does the rank grow in general? We initiate an answer to this question in this talk. This is joint work with Antonio Lei.

Oct 4

Renee Bell

Local-to-Global Extensions for Wildly Ramified Covers of Curves

Abstract: Given a Galois cover of curves $X \to Y$ with Galois group $G$ which is totally ramified at a point $x$ and unramified elsewhere, restriction to the punctured formal neighborhood of $x$ induces a Galois extension of Laurent series rings $k((u))/k((t))$. If we fix a base curve $Y$, we can ask when a Galois extension of Laurent series rings comes from a global cover of $Y$ in this way. Harbater proved that over a separably closed field, every Laurent series extension comes from a global cover for any base curve if $G$ is a $p$-group, and he gave a condition for the uniqueness of such an extension. Using a generalization of Artin--Schreier theory to non-abelian $p$-groups, we characterize the curves $Y$ for which this extension property holds and for which it is unique up to isomorphism, but over a more general ground field.

Oct 11

Chen Wan

A Local Trace Formula for the Generalized Shalika model

Abstract: I will discuss the local multiplicity problem for the generalized Shalika model. By proving a local trace formula for the model, we are able to prove a multiplicity formula for discrete series, which implies that the multiplicity of the generalized Shalika model is a constant over every discrete local Vogan L-packet. I will also discuss the relation between the multiplicity and the local exterior square L-function. This is a joint work with Rapheal Beuzart-Plessis.

Oct 18

Mark Shusterman

The fundamental group of a smooth projective curve over a finite field is finitely presented

Abstract: Solutions of (sets of) polynomial equations are (for quite some time) studied using the geometry of the associated varieties. The geometric approach was very successful, for instance, in the case of curves over finite fields. Associated to a curve is its (etale) fundamental group. This is a mysterious profinite group that ‘remembers’ the count of solutions to the equations giving rise to the curve, and sometimes also the curve itself (up to isomorphism). Grothendieck, using fundamental groups of complex curves, shed light on these mysterious profinite groups, showing (in particular) that they are finitely generated. We will show that these groups are furthermore finitely presented, hoping to find a finitary description for them (one day).

Oct 25

Douglas Ulmer

An algebraic approach to the Brauer-Siegel ratio for abelian varieties over function fields

https://www.dropbox.com/s/a5hjqgpn6joh033/seminar-abstract.pdf?dl=0

Nov 1

Jinbo Ren

Mathematical logic and its applications in number theory

Abstract: A large family of classical problems in number theory such as: a) Finding rational solutions of the so-called trigonometric Diophantine equation $F(\cos 2\pi x_i, \sin 2\pi x_i)=0$, where $F$ is an irreducible multivariate polynomial with rational coefficients; b) Determining all $\lambda \in \mathbb{C}$ such that $(2,\sqrt{2(2-\lambda)})$ and $(3, \sqrt{6(3-\lambda)})$ are both torsion points of the elliptic curve $y^2=x(x-1)(x-\lambda)$; c) Studying algebraicity of values of hypergeometric functions at algebraic numbers can be regarded as special cases of the Zilber-Pink conjecture in Diophantine geometry. In this talk, I will explain how we use tools from mathematical logic to attack this conjecture. In particular, I will present some partial results toward the Zilber-Pink conjecture, including those proved by Christopher Daw and myself.

Nov 8

Nick Andersen

Modular invariants for real quadratic fields

Abstract: The relationship between modular forms and quadratic fields is exceedingly rich. For instance, the Hilbert class field of an imaginary quadratic field may be generated by adjoining to the quadratic field a special value of the modular j-invariant. The connection between class groups of real quadratic fields and invariants of the modular group is much less understood. In my talk I will discuss some of what is known in this direction and present some new results (joint with W. Duke) about the asymptotic distribution of integrals of the j-invariant that are associated to ideal classes in a real quadratic field.

Nov 15

Ilya Khayutin

Equidistribution of Special Points on Shimura Varieties

Abstract: The André-Oort conjecture states that if a sequence of special points on a Shimura variety - Y - escapes all Hecke translates of proper Shimura subvarieties, viz. special subvarieties, then every irredicuble component of the Zariski closure of the sequence is an irreducible component of Y. A much stronger version of this conjecture is that the Galois orbits of a sequence of special points satisfying the assumption above equidistribute in connected components of Y. The latter conjecture would also imply the highly useful statement that the Galois orbits are dense in the analytic topology. Even more ambitiously, one would conjecture that orbits of large subgroups of the Galois group should equidistribute as well. The Pila-Zannier strategy which is the driving engine behind the spectacular recent progress on the André-Oort conjecture does not shed any light on these stronger questions of equidistribution and analytic density. The equidistribution conjecture is essentially known only for modular and Shimura curves following Duke’s pioneering result in the 80’s. I will discuss the relation of this problem to homogeneous dynamics and periodic torus orbits. I will then present two new theorems, for products of modular curves and for Kuga-Sato varieties, establishing partial results for the equidistribution conjecture by combining measure rigidity and a novel method to show that Galois/Torus orbits of special points do not concentrate on proper special subvarieties.

Nov 29

Valentijn Karemaker

Comparing obstructions to local-global principles for rational points over semiglobal fields

Abstract: Let K be a complete discretely valued field, let F be the function field of a curve over K, and let Z be a variety over F. When the existence of rational points on Z over a set of local field extensions of F implies the existence of rational points on Z over F, we say a local-global principle holds for Z. In this talk, we will compare local-global principles, and obstructions to such principles, for two choices of local field extensions of F. On the one hand we consider completions F_v at valuations of F, and on the other hand we consider fields F_P which are the fraction fields of completed local rings at points on the special fibre of a regular model of F. We show that if a local-global principle with respect to valuations holds, then so does a local-global principle with respect to points, for all models of F. Conversely, we prove that there exists a suitable model of F such that if a local-global principle with respect to points holds for this model, then so does a local-global principle with respect to valuations. This is joint work with David Harbater, Julia Hartmann, and Florian Pop.

Dec 6

Daniel Kriz

Title: A new p-adic Maass-Shimura operator and supersingular Rankin-Selberg p-adic L-functions

Abstract: We introduce a new p-adic Maass–Shimura operator acting on a space of “generalized p-adic modular forms” (extending Katz’s notion of p-adic modular forms), defined on the p-adic (preperfectoid) universal cover of a Shimura curve. Using this operator, we construct new p-adic L-functions in the style of Katz, Bertolini–Darmon–Prasanna and Liu–Zhang–Zhang for Rankin–Selberg families over imaginary quadratic fields K, in the ”supersingular” case where p is inert or ramified in K. We also establish new p-adic Waldspurger formulas, relating p-adic logarithms of elliptic units and Heegner points to special values of these p-adic L-functions. If time permits, we will discuss some applications to the arithmetic of abelian varieties.

Dec 13

Benjamin Bakker

Transcendence of period maps

Abstract: A period domain $D$ parametrizes Hodge structures and can be described as a certain analytic open set of a flag variety. Due to the presence of monodromy, the period map of a family of algebraic varieties lands in a quotient $D/\Gamma$ by an arithmetic group. In the very special case when $D/\Gamma$ is itself algebraic, understanding the interaction between algebraic structures on the source and target of the uniformization $D\rightarrow D/\Gamma$ is a crucial component of the modern approach to the Andr\'e-Oort conjecture. We prove a version of the Ax-Schanuel conjecture for general period maps $X\rightarrow D/\Gamma$ which says that atypical algebraic relations between $X$ and $D$ are governed by Hodge loci. We will also discuss some recent arithmetic applications due to Lawrence and Venkatesh. This is joint work with J. Tsimerman.