2020/12/21 by Naomi Sweeting, Sweeting, Naomi
Mathematics · Arts and Humanities · #Algebraic Geometry and Number Theory #Historical Studies and Socio-cultural Analysis #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2012.11771
Let E/ℚ be an elliptic curve and let K be an imaginary quadratic field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology classes for E using K-CM points and conjectured they did not all vanish. Conditional on this conjecture, he described the Selmer rank of E using his system of classes. We extend work of Wei Zhang to prove new cases of Kolyvagin's conjecture by considering congruences of modular forms modulo large powers of p . Additionally, we prove an analogous result, and give a description of the Selmer rank, in a complementary "definite" case (using certain modified L-values rather than CM points). Similar methods are also used to improve known results on the Heegner point main conjecture of Perrin-Riou. One consequence of our results is a new converse theorem, that p-Selmer rank one implies analytic rank one, when the residual representation has dihedral image.