Régulateurs syntomiques et valeurs de fonctions Lp-adiques I.
[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].
We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of a rank one quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.
In the study of the -adic sum of digits function , the arithmetical function and for plays a very important role. In this paper, we firstly generalize the relation between and to a bijective relation between arithmetical functions. And as an application, we investigate some aspects of the sum of digits functions induced by binary infinite Gray codes . We can show that the difference of the sum of digits function, , is realized by an automaton. And the summation formula of the sum...
For any positive power n of a prime p we find a complete set of generating relations between the elements [r] = rⁿ - r and p·1 of a unitary commutative ring.
We derive a relation between induced representations on the group which implies a relation between the jacobians of certain modular curves of level . The motivation for the construction of this relation is the determination of the applicability of Mazur’s method to the modular curve associated to the normalizer of a non-split Cartan subgroup of .