Displaying similar documents to “Class Invariants for Quartic CM Fields”

Remarks on strongly modular Jacobian surfaces

Xavier Guitart, Jordi Quer (2011)

Journal de Théorie des Nombres de Bordeaux

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In [] we introduced the concept of strongly modular abelian variety. This note contains some remarks and examples of this kind of varieties, especially for the case of Jacobian surfaces, that complement the results of [].

Torsion and Tamagawa numbers

Dino Lorenzini (2011)

Annales de l’institut Fourier

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Let K be a number field, and let A / K be an abelian variety. Let c denote the product of the Tamagawa numbers of A / K , and let A ( K ) tors denote the finite torsion subgroup of A ( K ) . The quotient c / | A ( K ) tors | is a factor appearing in the leading term of the L -function of A / K in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over ℚ or quadratic extensions K / ℚ , and for abelian surfaces A / ℚ . The smallest possible...

Selmer groups for elliptic curves in ℤ l d -extensions of function fields of characteristic p

Andrea Bandini, Ignazio Longhi (2009)

Annales de l’institut Fourier

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Let F be a function field of characteristic p > 0 , ℱ / F a ℤ l d -extension (for some prime l ≠ p ) and E / F a non-isotrivial elliptic curve. We study the behaviour of the r -parts of the Selmer groups ( r any prime) in the subextensions of ℱ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of ℱ / F .

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

Matteo Longo (2006)

Annales de l’institut Fourier

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Let E / F be a modular elliptic curve defined over a totally real number field F and let φ be its associated eigenform. This paper presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of E over suitable quadratic imaginary extensions K / F . In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when [ F : ℚ ] is even and φ not new at any prime.