TBA

Let $G$ be a finitely presented residually finite group. We are interested in the growth of size of the torsion of $H^{ab}$ as a function of $|G:H|$ where $H$ ranges over normal subgroups of finite index in $G$. It is easy to see that this grows at most exponentially in terms of $|G:H|$. Of particular interest is the case when $G$ is an arithmetic hyperbolic 3-manifold group and $H$ ranges over its congruence subgroups. Proving exponential lower bounds on the torsion appears to be difficult and in this talk I will focus on the situation of finitely presented pro-$p$ groups.

In contrast with abstract groups I will show that in finitely presented pro-$p$ groups torsion in the abelianizations can grow arbitrarily fast. The examples are rather 'large' pro-$p$ groups, in particular they are virtually Golod-Shafarevich. When we restrict to $p$-adic analytic groups the torsion growth is at most polynomial.

The Jacquet-Langlands correspondence gives a relationship between automorphic representations on $GL_2$ and its twisted forms, which are the unit groups of quaternion algebras. Writing this out in more classical language gives a combinatorial way of producing the eigenvalues of Hecke operators acting on modular forms. In this talk, we will first go over notions of modular forms and quaternion algebras, and then dive into an explicit example by computing some eigenvalues of the lowest level quaternionic modular form of weight $2$ over $\mathbb{Q}$.

We can define a continued fraction for formal series $f(t)=\sum_{i=-\infty}^d c_it^i$ by repeatedly removing the polynomial part, $\sum_{i=0}^d c_it^i$, (the equivalent of the integer part) and inverting the remaining part, as in the real case. This way, the partial quotients are polynomials. Both the usual continued fractions and the polynomial continued fractions carry properties of best approximation. However, while for square roots of rationals the real continued fraction is eventually periodic, such periodicity does not always occur for $\sqrt{D(t)}$. The correct analogy was found by Abel in 1826: the continued fraction of $\sqrt{D(t)}$ is eventually periodic if and only if there exist nontrivial polynomials $x(t)$, $y(t)$ such that $x(t)^2-D(t)y(t)^2=1$ (the polynomial Pell equation). Notice that the same holds also for square root of integers in the real case. In 2014 Zannier found that some periodicity survives for all the $\sqrt{D(t)}$: the degrees of their partial quotients are eventually periodic. His proof is strongly geometric and it is based on the study of the Jacobian of the curve $u^2=D(t)$. We give a brief survey of the theory of polynomial continued fractions, Jacobians and an account of the proof of the result of Zannier.