Displaying similar documents to “On a conjecture of Watkins”

On uniform lower bound of the Galois images associated to elliptic curves

Keisuke Arai (2008)

Journal de Théorie des Nombres de Bordeaux

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Let p be a prime and let K be a number field. Let ρ E , p : G K → Aut ( T p E ) ≅ GL 2 ( ℤ p ) be the Galois representation given by the Galois action on the p -adic Tate module of an elliptic curve E over K . Serre showed that the image of ρ E , p is open if E has no complex multiplication. For an elliptic curve E over K whose j -invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of ρ E , p .

Signed Selmer groups over p -adic Lie extensions

Antonio Lei, Sarah Livia Zerbes (2012)

Journal de Théorie des Nombres de Bordeaux

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Let E be an elliptic curve over ℚ with good supersingular reduction at a prime p ≥ 3 and a p = 0 . We generalise the definition of Kobayashi’s plus/minus Selmer groups over ℚ ( μ p ∞ ) to p -adic Lie extensions K ∞ of ℚ containing ℚ ( μ p ∞ ) , using the theory of ( ϕ , Γ ) -modules and Berger’s comparison isomorphisms. We show that these Selmer groups can be equally described using Kobayashi’s conditions via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer groups in the...

The integral logarithm in Iwasawa theory : an exercise

Jürgen Ritter, Alfred Weiss (2010)

Journal de Théorie des Nombres de Bordeaux

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Let l be an odd prime number and H a finite abelian l -group. We describe the unit group of Λ ∧ [ H ] (the completion of the localization at l of ℤ l [ [ T ] ] [ H ] ) as well as the kernel and cokernel of the integral logarithm L : Λ ∧ [ H ] × → Λ ∧ [ H ] , which appears in non-commutative Iwasawa theory.

CM liftings of supersingular elliptic curves

Ben Kane (2009)

Journal de Théorie des Nombres de Bordeaux

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Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D < 0 such that the reduction map modulo a prime above p from elliptic curves with CM by 𝒪 D to supersingular elliptic curves in characteristic p is surjective. In the algorithm we first determine an explicit constant D p so that | D | > D p implies that the map is necessarily surjective and then we compute explicitly the cases | D | < D p .