# Finding Large Selmer Rank via an Arithmetic Theory of Local Constants

 Title: Finding Large Selmer Rank via an Arithmetic Theory of Local Constants Author: Mazur, Barry C.; Rubin, Karl Note: Order does not necessarily reflect citation order of authors. Citation: Mazur, Barry C., and Karl Rubin. 2007. Finding large Selmer rank via an arithmetic theory of local constants. Annals of Mathematics 166(2): 579-612. Access Status: Full text of the requested work is not available in DASH at this time (“dark deposit”). For more information on dark deposits, see our FAQ. Full Text & Related Files: Mazur_FindSelmerRank.pdf (310.1Kb; PDF) Abstract: We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose $$K∕k$$ is a quadratic extension of number fields, $$E$$ is an elliptic curve defined over k,and p is an odd prime. Let $$K−$$ denote the maximal abelian p-extension of $$K$$ that is unramified at all primes where E has bad reduction and that is Galois over $$k$$ with dihedral Galois group (i.e., the generator c of Gal$$(K∕k)$$ acts on $$Gal(K−∕K)$$ by inversion). We prove (under mild hypotheses on $$p$$) that if the $$Zp$$-rank of the pro-$$p$$ Selmer group $$S_p(E∕K)$$ is odd, then $$rankZ_p S_p(E∕F) \ge [F : K]$$ for every finite extension $$F$$ of $$K$$ in $$K−$$. Published Version: doi:10.4007/annals.2007.166.579 Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10355839 Downloads of this work: