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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...

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