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It is known that, for every elliptic curve over ℚ, there exists a quadratic extension in which the rank does not go up. For a large class of elliptic curves, the same is known with the rank replaced by the size of the 2-Selmer group. We show, however, that there exists a large supply of semistable elliptic curves  E /ℚ whose 2-Selmer group grows in size in every bi-quadratic extension, and such that, moreover, for any odd prime  p , the size of the  p -Selmer group grows in every  D  2 p  -extension and every elementary abelian  p -extension of rank at least 2. We provide a simple criterion for an elliptic curve over an arbitrary number field to exhibit this behaviour. We also discuss generalizations to other Galois groups.

ELLIPTIC CURVES WITH p -SELMER GROWTH FOR ALL p