NTS Spring 2015 Abstract
|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.
|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.
|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.
|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.
|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.
|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.
|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.
|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.
Organizer contact information
Sean Rostami (email@example.com)
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