BOUNDS FOR 2-SELMER RANKS IN TERMS OF SEMINARROW CLASS GROUPS

Let E be an elliptic curve over a number field K defined by a monic irreducible cubic polynomial F(x). When E is nice at all finite primes of K, we bound its 2-Selmer rank in terms of the 2-rank of a modified ideal class group of the field L = K[x]/(F(x)), which we call the seminarrow class group of L. We then provide several sufficient conditions for E being nice at a finite prime.As an application, when K is a real quadratic field, E/K is semistable and the discriminant of F is totally negative, we frequently determine the 2-Selmer rank of E by computing the root number of E and the 2-rank of the narrow class group of L.