Displaying similar documents to “Torsion and Tamagawa numbers”

On rational torsion points of central ℚ -curves

Fumio Sairaiji, Takuya Yamauchi (2008)

Journal de Théorie des Nombres de Bordeaux

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Let E be a central ℚ -curve over a polyquadratic field k . In this article we give an upper bound for prime divisors of the order of the k -rational torsion subgroup E t o r s ( k ) (see Theorems 1.1 and 1.2). The notion of central ℚ -curves is a generalization of that of elliptic curves over ℚ . Our result is a generalization of Theorem 2 of Mazur [], and it is a precision of the upper bounds of Merel [] and Oesterlé [].

Some remarks on almost rational torsion points

John Boxall, David Grant (2006)

Journal de Théorie des Nombres de Bordeaux

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For a commutative algebraic group G over a perfect field k , Ribet defined the set of almost rational torsion points G tors , k ar of G over k . For positive integers d , g , we show there is an integer U d , g such that for all tori T of dimension at most d over number fields of degree at most g , T tors , k ar ⊆ T [ U d , g ] . We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit...

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.

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 .