Multiplicities in Selmer groups and root numbers of Artin twists

Let K / F be a finite Galois extension of number fields and let σ an absolutely irreducible, self-dual, complex valued representation Gal ( ) . p odd prime consider two elliptic curves E 1 , 2 defined over Q with good, ordinary reduction at primes above equivalent mod- representations. In this article, we study the variation parity multiplicities in space associated to ∞ -Selmer groups We also compare root numbers for twists by show that -parity conjecture holds twist if only it express Mazur-Rubin-Nekovář's arithmetic local constants terms certain Iwasawa invariants.