Talk: Level Raising Mod 2 and Arbitrary 2-selmer Ranks
نویسنده
چکیده
rankE(Q) ? = ords=1 L(E/Q, s). This rank conjecture is known when ords=1 L(E/Q, s) ≤ 1 (Gross-Zagier, Kolyvagin, ...). The recent breakthrough of Bhargava-Skinner-W. Zhang shows that the rank conjecture holds for at least 66% of all elliptic curves over Q. Even more interestingly, BSD further predicts a refined BSD formula for the leading term of L(E/Q, s) at s = 1 in terms of various important arithmetic invariants of E:
منابع مشابه
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...
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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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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 ...
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In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find m...
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تاریخ انتشار 2015