Level Raising Mod 2 and Obstruction to Rank Lowering
نویسنده
چکیده
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 parity, as predicted by the BSD conjecture. When the 2-Selmer rank of E is one, we further prove that the 2-Selmer rank of A can never be zero, revealing a surprising obstruction to rank lowering which is unseen for p-Selmer groups for odd p.
منابع مشابه
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....
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