On the Bloch-Kato conjecture for $K_2$ of some elliptic curves, and some indivisibility results

par Prof. Rob de Jeu (Vrije Universiteit Amsterdam)

vendredi 16 janv. 2026, 15:00 → 16:30 Europe/Paris

Salle Olga Ladyjenskaïa (IHP)

For a number field $F$, the classical expression for the residue at $s=1$ of its zeta-function $\zeta_F(s)$ involves the regulator of the units of its ring of algebraic integers as well as its class number. In the 1970s, Borel for $n\ge2$ proved a similar relation between a regulator of $K_{2n-1}(F)$ and $\zeta_F(n)$, and Bloch for certain elliptic curves $E$ over the rationals proved such a relation between $L(E,2)$ and a regulator of an element of $K_2(E)$. Such relations involve rational numbers, which, according to a later conjecture by Bloch and Kato, should be interpreted using $p$-adic regulator maps.

After reviewing this history and background, we discuss some calculations and results towards this Bloch-Kato conjecture for $p=2$, for a certain family of elliptic curves over the rationals.  In the process, we prove some indivisibility results, both for some elements in $K$ groups and for their images under the $2$-adic regulator map.

This is joint work with Neil Dummigan, Vasily Golyshev and Matt Kerr.