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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'(6) of level 6 defined over whose -rational points correspond to j-invariants of elliptic curves E over for which (E[2]) is a subfield of (E[3]). They characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model of X'(6). X'(6)() 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 .