An official website of the United States government
Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to non-federal websites. Their policies may differ from this site.
-
We show that for certain non-CM elliptic curves E / Q E_{/\mathbb {Q}} such that 3 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E Ï E_{\psi } of E E have MordellâWeil rank one and the 3 3 -adic height pairing on E Ï ( Q ) E_{\psi }(\mathbb {Q}) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p p -adic height. It is not known â though expected â that the archimedian height of these higher-codimensional cycles is non-zero.more » « less
-
RĂ©sumĂ© Soit $$E/{\mathbb {Q}}$$ E / Q une courbe elliptique Ă multiplication complexe et p un nombre premier de bonne rĂ©duction ordinaire pour E . Nous montrons que si $${\mathrm{corank}}_{{\mathbb {Z}}_p}{\mathrm{Sel}}_{p^\infty }(E/{\mathbb {Q}})=1$$ corank Z p Sel p â ( E / Q ) = 1 , alors E a un point dâordre infini. Le point de non-torsion provient dâun point de Heegner, et donc $${{{\mathrm{ord}}}}_{s=1}L(E,s)=1$$ ord s = 1 L ( E , s ) = 1 , ce qui donne une rĂ©ciproque Ă un thĂ©orĂšme de GrossâZagier, Kolyvagin, et Rubin dans lâesprit de [49, 54]. Pour $$p>3$$ p > 3 , cela donne une nouvelle preuve du rĂ©sultat principal de [12], que notre approche Ă©tend Ă tous les nombres premiers. Lâapproche se gĂ©nĂ©ralise aux courbes elliptiques Ă multiplication complexe sur les corps totalement rĂ©els [4].more » « less
-
Abstract We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of $${{\,\mathrm{GL}\,}}_3$$ GL 3 over imaginary quadratic fields, using the cohomology of Shimura varieties for $${\text {GU}}(2, 1)$$ GU ( 2 , 1 ) .more » « less
