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On a theorem of Mestre and Schoof

John E. Cremona, Andrew V. Sutherland (2010)

Journal de Théorie des Nombres de Bordeaux

A well known theorem of Mestre and Schoof implies that the order of an elliptic curve E over a prime field 𝔽 q can be uniquely determined by computing the orders of a few points on E and its quadratic twist, provided that q > 229 . We extend this result to all finite fields with q > 49 , and all prime fields with q > 29 .

On the classification of 3-dimensional non-associative division algebras over p -adic fields

Abdulaziz Deajim, David Grant (2011)

Journal de Théorie des Nombres de Bordeaux

Let p be a prime and K a p -adic field (a finite extension of the field of p -adic numbers p ). We employ the main results in [12] and the arithmetic of elliptic curves over K to reduce the problem of classifying 3-dimensional non-associative division algebras (up to isotopy) over K to the classification of ternary cubic forms H over K (up to equivalence) with no non-trivial zeros over K . We give an explicit solution to the latter problem, which we then relate to the reduction type of the jacobian...

On the de Rham and p -adic realizations of the elliptic polylogarithm for CM elliptic curves

Kenichi Bannai, Shinichi Kobayashi, Takeshi Tsuji (2010)

Annales scientifiques de l'École Normale Supérieure

In this paper, we give an explicit description of the de Rham and p -adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve E defined over an imaginary quadratic field 𝕂 with complex multiplication by the full ring of integers 𝒪 𝕂 of 𝕂 . Note that our condition implies that 𝕂 has class number one. Assume in addition that E has good reduction above a prime p 5 unramified in 𝒪 𝕂 . In this case, we prove that the specializations of the p -adic elliptic...

R -équivalence sur les familles de variétés rationnelles et méthode de la descente

Alena Pirutka (2012)

Journal de Théorie des Nombres de Bordeaux

La méthode de la descente a été introduite et développée par Colliot-Thélène et Sansuc. Elle permet d’étudier l’arithmétique de certaines variétés rationnelles. Dans ce texte on montre comment il en résulte que pour certaines familles f : X Y de variétés rationnelles sur un corps local k de caractéristique nulle le nombre des classes de R -équivalence de la fibre X y ( k ) est localement constant quand y varie dans Y ( k ) .

Refined theorems of the Birch and Swinnerton-Dyer type

Ki-Seng Tan (1995)

Annales de l'institut Fourier

In this paper, we generalize the context of the Mazur-Tate conjecture and sharpen, in a certain way, the statement of the conjecture. Our main result will be to establish the truth of a part of these new sharpened conjectures, provided that one assume the truth of the classical Birch and Swinnerton-Dyer conjectures. This is particularly striking in the function field case, where these results can be viewed as being a refinement of the earlier work of Tate and Milne.

Signed Selmer groups over p -adic Lie extensions

Antonio Lei, Sarah Livia Zerbes (2012)

Journal de Théorie des Nombres de Bordeaux

Let E be an elliptic curve over 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 ( μ p ) to p -adic Lie extensions K of containing ( 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 ordinary case....

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