On the Parity of Ranks of Selmer Groups III

Ja n 20 01 On the parity of ranks of Selmer groups

Level Raising Mod 2 and Obstruction to Rank Lowering

Given an elliptic curve E defined over Q, we are motivated by the 2-part of the Birch and Swinnerton-Dyer formula to study the relation between the 2-Selmer rank of E and the 2-Selmer rank of an abelian variety A obtained by Ribet’s level raising theorem. For certain imaginary quadratic fields K satisfying the Heegner hypothesis, we prove that the 2-Selmer ranks of E and A over K have different...

Disparity in Selmer ranks of quadratic twists of elliptic curves

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example ...

Level Raising Mod 2 and Obstruction for Rank Lowering

Given an elliptic curve E defined over Q, we are motivated by the 2-part of the Birch and Swinnerton-Dyer formula to study the relation between the 2-Selmer rank of E and the 2-Selmer rank of an abelian variety A. This abelian variety A is associated to a modular form g of weight 2 and level Nq that is obtained by Ribet’s level raising theorem from the modular form f of level N associated to E....

The Growth of Selmer Ranks of an Abelian Variety with Complex Multiplication

Let K be a CM field and O be its ring of integers. Let p be an odd prime integer and p be a prime in K lying above p. Let F be a Galois extension of K unramified over p. For an Abelian variety A defined over F with complex multiplication by O, we study the variation of the p-ranks of the Selmer groups in pro-p algebraic extensions. We first study the Zpextension case. When K is quadratic imagin...