Through the universal $G$-space $EG$, once can replace a $G$-space $X$ by a homotopy equivalent space in which the $G$-action is free. the orbit space of this action is the Borel construction. This Borel construction is then a better model to study the $G$-action, as not only can we use tools of algebraic topology such as cohomology (which leads to equivariant cohomology) and spectral sequences, but also computations using these tools are often way nicer.

Meeting ID: 997 4840 9440 Passcode: 911104

We will discuss how to compute dynamical invariants of a surjective endomorphism of a projective bundle over an elliptic curve. This has applications to arithmetic dynamics and the Kawaguchi-Silverman conjecture. In particular, we will discuss how to use the theory of Schur functors to compute the Iitaka dimension of certain projective bundles. This extends work done by Atiyah in classifying vector bundles on elliptic curves.

Given a domain $\Omega\subseteq\mathbb{C}^n$, the space of square-integrable holomorphic functions on $\Omega$ is a Hilbert space with the standard inner product. This space is denoted by $L^2_h(\Omega)$ and is known as the Bergman space of $\Omega$. It can be shown that the evaluation functionals $E_z:L^2_h(\Omega)\to\mathbb{C}$ given by $E_z(f)=f(z)$ are continuous on $L^2_h(\Omega)$, and hence via the Riesz representation theorem there exists a $K(\,\cdot\,,z)\in L^2_h(\Omega)$ that reproduces square-integrable holomorphic functions on $\Omega$. This function (on $\Omega\times\Omega$) is known as the Bergman kernel of $\Omega$, and has had a profound impact on the theory of holomorphic functions of several complex variables. This theory can also be generalized to weighted $L^2$-spaces, given that the weight function is sufficiently "nice". $$ $$ This will be a mostly expository talk on Bergman kernel, with an emphasis on weighted Bergman kernels. Time allowing I will sketch some work I have done regarding the zeroes of weighted Bergman kernels.

Let $E$ be an elliptic curve with non-constant $j$-invariant over a function field $K$ with constant field of size an odd prime power $q$. Its $L$-function $L(T,E/K)$ belongs to $1 + T\mathbb{Z}[T]$. Inspired by the algorithms of Schoof and Pila for computing zeta functions of curves over finite fields, we propose an approach to compute $L(T,E/K)$. The idea is to compute, for sufficiently many primes $\ell$ invertible in $K$, the reduction $L(T,E/K) \bmod{\ell}$. The $L$-function is then recovered via the Chinese remainder theorem. When $E(K)$ has a subgroup of order $N \geq 2$ coprime with $q$, Chris Hall showed how to explicitly calculate $L(T,E/K) \bmod{N}$. We present novel theorems going beyond Hall's.