## Section: New Results

### Number and function fields

Participants : Jean-Marc Couveignes, Karim Belabas.

In the article [29] , J. Brau study the growth of the Galois invariants of the $p$-Selmer group of an elliptic curve in a degree $p$ Galois extension. He shows that this growth is determined by certain local cohomology groups and determine necessary and sufficient conditions for these groups to be trivial.

In the article [30] written with J. Nathan, J. Brau study the modular curve ${X}^{\text{'}}\left(6\right)$ of level 6 defined over $\mathbb{Q}$ whose $\mathbb{Q}$-rational points correspond to $j$-invariants of elliptic curves $E$ over $\mathbb{Q}$ for which $\mathbb{Q}\left(E\right[2\left]\right)$ is a subfield of $\mathbb{Q}\left(E\right[3\left]\right)$. They characterize the $j$-invariants of elliptic curves with this property by exhibiting an explicit model of ${X}^{\text{'}}\left(6\right)$. ${X}^{\text{'}}\left(6\right)\left(\mathbb{Q}\right)$ gives an infinite family of examples of elliptic curves with non-abelian "entanglement fields," which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over $\mathbb{Q}$.