# Algebra and Number Theory Seminar Spring 2019

**Friday April 12th, 2019**

**Florian Sprung, Arizona State University**

*How does the rank of an elliptic curve grow in towers of number fields?*

On an elliptic curve *y ^{2} = x^{3} + ax + b*, the points with coordinates (

*x, y*) in a given number field form a finitely generated abelian group. One natural question is how the rank of this group changes when changing the number field. For the simplest example with infinitely many number fields, fix a prime p. Adjoining to Q the

*p*th,

*p*

^{2}th,

*p*

^{3}th,... roots of unity produces a tower of number fields

**Q ⊂ Q**(ζp) ⊂** Q**(ζp^{2} )** ⊂ ....**

One may guess that the rank should keep growing in this tower (’more numbers mean more solutions’). However, this guess turns out to be incorrect – the rank is always bounded, as envisioned by the theories of Iwasawa and Mazur in the 1970’s. The above tower started with Q, but there are analogous towers that start with an imaginary quadratic field instead. Given the above boundedness result, one would now guess that the rank is bounded in these towers, too. Surprisingly, this is not the case – there are scenarios both for bounded and unbounded rank. So how does the rank grow in those towers in general? We initiate an answer to this question in this talk. This is joint work with Antonio Lei.

**Friday April 19th, 2019**

**TBA**

**Friday April 26th, 2019**

**TBA**

**Friday May 3rd, 2019**

**Chris Marks, Chico State University**

*Period relations for Riemann surfaces with many automorphisms*

Algebraic curves whose associated Jacobian variety has complex multiplication (CM) have long been recognized as important objects in arithmetic geometry and algebraic number theory, as for example such curves feature prominently both in Hilbert’s 12th problem regarding explicit class field theory and also more recently in Stark’s conjecture regarding certain units in number fields. Such varieties occur when the periods of the underlying curve have an abnormally high number of linear relations over the algebraic numbers, but in general both the computation of these periods and the determination of the number of such relations are extremely difficult problems. In this talk, I will explain some ongoing research with Luca Candelori from Wayne State University, which has greatly improved the previous best estimate for the number of relations among periods for algebraic curves with “many automorphisms”, and time permitting I will give a heuristic that might allow one to verify (or disprove) Coleman’s conjecture regarding the number of CM curves of a given genus.

**Friday May 10th, 2019**

**TBA**

**Friday May 17th, 2019**

**Richard Gottesman, Queen's University**

*Vector-Valued Modular Forms*

Vector-valued modular forms are a natural generalization of classical modular forms. Their study is important in number theory, geometry, and the theory of vertex operator algebras. The collection of all vector-valued modular forms form a graded module over the ring of classical modular forms. I will explain how understanding the structure of this module (it is always Cohen-Macaulay!) allows us to obtain explicit formulas for the Fourier coefficients of vector-valued modular forms. A key application is certain cases of the unbounded denominator conjecture for vector-valued modular forms. No previous knowledge of vector-valued modular forms will be assumed.

**Friday May 24th, 2019**

**Paul VanKoughnett, Purdue University**

*A new approach to Goerss-Hopkins obstruction theory*

Let E be a homology theory and M an E_*E-comodule. Is there a spectrum X such that M is the E-homology of X, and if so, how many? Goerss-Hopkins obstruction theory is a tool for answering such questions, including analogous ones for highly structured ring spectra. In joint work with Piotr Pstrągowski, we introduced a general infinity-categorical framework for obstruction theory using synthetic spectra. We'll set up this framework, observe it in action, and discuss the relationship between convergence of E-based obstruction theories and the homological algebra of E_*E-comodules.

**Friday May 31st, 2019**

**Suzana Milea, University of California Santa Cruz**

**Friday June 7th, 2019**

**TBA**