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It is known, that for every elliptic curve over Q there exists a quadraticextension in which the rank does not go up. For a large class of ellipticcurves, the same is known with the rank replaced by the 2-Selmer group. Weshow, however, that there exists a large supply of semistable elliptic curvesE/Q whose 2-Selmer group goes up in every bi-quadratic extension and for anyodd prime p, the p-Selmer group goes up in every D_{2p}-extension and everyelementary abelian p-extension of rank at least 2. We provide a simplecriterion for an elliptic curve over an arbitrary number field to exhibit thisbehaviour. We also discuss generalisations to other Galois groups.

Elliptic curves with p-Selmer growth for all p