# NTS Spring 2015 Abstract

## Jan 29

 Lillian Pierce Averages and moments associated to class numbers of imaginary quadratic fields Given an imaginary quadratic field of discriminant d, consider the p-part of the associated class number for a prime p. This quantity is well understood for p=2, and significant results are known for p=3, but much less is known for larger primes. One important type of question is to prove upper bounds for the p-part. Desirable upper bounds could take several forms: either “pointwise” upper bounds that hold for the p-part uniformly over all discriminants, or upper bounds for the p-part when averaged over all discriminants, or upper bounds for higher moments of the p-part. This talk will discuss recent results (joint work with Roger Heath-Brown) that provide new upper bounds for averages and moments of p-parts for odd primes.

## Feb 05

 Keerthi Madapusi Heights of special divisors on orthogonal Shimura varieties The Gross-Zagier formula relates two complex numbers obtained in seemingly very disparate ways: The Neron-Tate height pairing between Heegner points on elliptic curves, and the central derivative of a certain automorphic L-function of Rankin type. I will explain a variant of this in higher dimensions. On the geometric side, the intersection theory will now take place on Shimura varieties associated with orthogonal groups. On the analytic side, we will find Rankin-Selberg L-functions involving modular forms of half-integral weight. This is joint work with Fabrizio Andreatta, Eyal Goren and Ben Howard.

## Feb 19

 David Zureick-Brown The canonical ring of a stacky curve We give a generalization to stacks of the classical theorem of Petri -- i.e., we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. (The talk will be mostly geometric and will require little understanding of modular forms.) This is joint work with John Voight.

## Feb 26

 Rachel Davis Origami and Galois representations Let E be an elliptic curve over Q. An origami is a pair (C, f), where C is a curve and f : C → E is a map, branched only above one point. When C is E and f is multiplication by n, there is an associated Galois representation (that depends on a rational point P ∈ E) to an affine general linear group. We explain this and then study the Galois theory for origami with non-abelian monodromy groups. This is joint work with Edray Goins.

## Mar 05

 Hongbo Yin Some non-congruence subgroups and the associated modular curves We give some new non-congruence subgroups which is close to the Fermat groups but with very different properties.

## Mar 12

 Tonghai Yang Informal tutorial on the Heisenberg group and Weil representation This will be an informal tutorial on the Heisenberg group and the Weil representation of symplectic groups over local fields and, if time permits, adele rings. The motivation is partly via theta functions.

## Mar 19

 Brian Cook Forms in many variables and subsets of the integers This talk is about problems related to forms in many variables where the variables are restricted to certain subsets of the integers.

## Apr 09

 Daniel Ross Boston-Bush-Hajir heuristics: Moments and function fields Cohen and Lenstra developed a heuristic predicting the distribution of the p-parts of the class groups of imaginary quadratic number fields--that is, the Galois groups of the maximal unramified abelian p-extensions of the imaginary quadratics. Boston, Bush, and Hajir extended this to a heuristic describing the distribution of the maximal unramified (no longer necessarily abelian) p-extensions of the imaginary quadratics. In this talk we explain this distribution its rephrasing in terms of its moments, and some evidence for the conjecture coming from its function field analogue.

## Apr 16

 Nigel Boston Beauville Surfaces and Groups Interest in constructing exotic algebraic varieties goes back to examples of David Mumford and Armand Beauville in the late seventies. In the early 2000s, a systematic study of Beauville surfaces was initiated in works of Bauer, Catanese and Grunewald, and turned out to be connected to many subjects. I will survey what is known about Beauville surfaces and related groups and present a first infinite family of so-called mixed Beauville surfaces using groups acting on buildings, obtained by Barker, Peyerimhoff, Vdovina, and me.

## Apr 23

 Filip Najman Mordell-Weil groups of elliptic curves over number fields The Mordell-Weil group E(K) of K-rational points of an elliptic curve E over a number field K, is a finitely generated abelian group and hence isomorphic to the direct product of its torsion subgroup and Zr, where r is the rank of E/K. In this talk we will consider the question of what this group can be over number fields of certain type, e.g. over all number fields of degree d or over a fixed number field. After surveying known results, both old and new, about torsion groups, we will show that prescribing the torsion over number fields (as opposed to over Q!) can force various properties on the elliptic curve. For instance, all elliptic curves with points of order 13 or 18 over quadratic fields have to have even rank and elliptic curves with points of order 16 over quadratic fields are base changes of elliptic curves defined over Q. We show that these properties arise from the geometry of the corresponding modular curves.

## Apr 30

 David Roberts Hurwitz Belyi Maps A degree n covering map from a Riemann surface to the Riemann sphere generically has many critical values in the base Riemann sphere, each of multiplicity one. At the other extreme, the map is called a Belyi map if its critical value set is in {0, 1, ∞}. Belyi maps are very interesting arithmetically. For example, they are automatically defined over Q and have bad reduction only at primes ≤ n. I will talk about particularly interesting Belyi maps arising from solutions to Hurwitz moduli problems. Typically they are defined over Q and in my examples they will have bad reduction at three primes only, despite their arbitrarily large degree. A focus will be on methods of computation and explicit examples.

## May 07

 Jesse Wolfson Geometry and Arithmetic of Spaces of Rational Maps Topological relationships between algebraic varieties should have algebraic explanations. In this talk, I will report on ongoing joint work with Benson Farb on the geometry and arithmetic of spaces of rational maps and monic polynomials. By combining topological arguments of Segal with work of Bjorner-Ekedahl, we compute the weight filtration on the $\ell$-adic cohomology of spaces of rational maps and of monic polynomials. As a consequence, we obtain algebro-geometric refinements of some of the topological relationships between these spaces discovered by Cohen-Cohen-Mann-Milgram, Vassiliev and others.

## May 14

 Quoc Ho Average Size of 2-Selmer Groups of Elliptic Curves over Function Fields Employing a geometric setting inspired by the proof of the Fundamental Lemma, we study some counting problems related to the average size of 2-Selmer groups and hence obtain an estimate for it. This joint work with B. Le Hung and B.C Ngo.