On the parity ranks of Selmer groups

Assume that p > 3 and that / is ordinary at p, i.e. that ap(f) 6 Fp is a p-adic unit. According to Hida's theory, there is a p-adic family of ordinary modular forms of varying weights containing / (we ignore the phenomenon of "p-stabilization" in this Introduction). In concrete terms, this means that there is an integer c > 0 such that for every integer k > 2 satisfying k = ko (mod (p — l)p), there is an ordinary newform fk of weight k on ro(iV) such that /fc0 = / and . k = k' (mod (p l)p) implies /* = /*' (modp").

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