Analytic curves in algebraic varieties over number fields

We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of Borel-Dwork and P\'olya-Bertrandias valid over the projective line to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and rigid analytic geometry (notably, potential theory over complex and $p$-adic curves). We also discuss geometric analogues, pertaining to the algebraic geometry of projective surfaces, of these arithmetic criteria.Comment: 55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkh\"auser, 200

A p-adic analogue of Siegel's Theorem on sums of squares

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p ‐adic Kochen operator provides a p ‐adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p ‐integral elements of K . We use this to formulate and prove a p ‐adic analogue of Siegel's theorem, by introducing the p ‐Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p ‐Pythagoras number and show that the growth of the p ‐Pythagoras number in finite extensions is bounded

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2, three references added ([Mas75], [MB90] and [Yu94]) and one reference updated [BS12]. Accepted in Bull. Brazil. Mat. So

Correspondences in Arakelov geometry and applications to the case of Hecke operators on modular curves

In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due independently to Bost and Kuehn we compute these invariants in terms of special values of L series. On the other hand, we obtain a proof of the fact that Hecke correspondences acting on the Jacobian of the modular curves are self-adjoint with respect to the N\'eron-Tate height pairing.Comment: 38 pages. Minor correction

GEOMETRICAL SYMMETRIES IN SPHERICAL TOP AND OTHER MOLECULES

Author Institution: Laboratoire de Spectronomie Mol\'{e}culaire, Universit\'{e} de Dijo

ROTATION-VIBRATION ENERGY LEVELS OF TETRAHEDRAL XY4XY_{4} MOLECULES IN A DEGENERATE VIBRATIONAL STATE v=1v = 1, II. THREEFOLD DEGENERATE VIBRATIONS (v3(v_{3} OR v4=1v_{4} = 1)

Author Institution: Facult\'e des SciencesThe case of a threefold degenerate vibrational state is more complex due to the Coriolis interaction between the degenerate vibration and the rotation; important contributions were made to this problem by II. A. Jahn, J. D. Louck and K. T. Hecht, using symmetry considerations. However new secular equations had to be solved, each time a new band or a new molecule was studied. We propose a method where the secular equations can be solved numerically once for all. It is then possible to use explicit formulae expressing the wave numbers of the lines as functions of quantum numbers and molecular parameters. This method has been applied to the fundamental $\nu_{4}$ of methane

La stricte incompatibilité entre la qualité d’expert en automobile et celle d’assureur

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La stricte incompatibilité entre la qualité d’expert en automobile et celle d’assureur

Dossier « Expertise automobile, les 50 ans de la loi fondatrice »National audienc

Qu'est-ce que la déontologie ?

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