Talk

After recalling the celebrated formula by Gross and Zagier relating the derivative of the Hasse-Weil L-function of elliptic curves to Heegner points, I explain its p-adic analog and an application for the Birch and Swinnerton-Dyer conjecture. I will also overview the Iwasawa theory of elliptic curves, which plays a crucial role in the application.

The Euler-Kronecker constant of a number field $K$ is the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_K(s)$ at $s=1$. We study the distribution of the Euler-Kronecker constant $\gamma_q^+$ of the maximal real subfield of $\mathbb Q(\zeta_q)$ as $q$ ranges over the primes. Further, we consider the distribution of $\gamma_q^+-\gamma_q$, with $\gamma_q$ the Euler--Kronecker constant of $\mathbb Q(\zeta_q)$ and show how it is connected with Kummer's conjecture.

Two CW-complexes are said to be simple homotopy equivalent if they are related by a sequence of collapses and expansions of cells. This notion interpolates between homeomorphism and homotopy in the sense that simple homotopy equivalent implies homotopy equivalent, and homeomorphic implies simple homotopy equivalent. The aim of this talk will be to present the first examples of two 4-manifolds which are homotopy equivalent but not simple homotopy equivalent, as well as in all higher even dimensions. This is joint work with Csaba Nagy and Mark Powell (arXiv:2312.00322).

The talk plans to present an introduction to the eponymous conjecture predicting a mysterious link between rational points on an elliptic curve defined over rational numbers,
and analytic properties of the associated Hasse-Weil L-function. Some recent progress will be discussed.