# Difference between revisions of "NTS ABSTRACTFall2018"

## 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 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.