38 research outputs found
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 âadic Kochen operator provides a p âadic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p âintegral elements of K . We use this to formulate and prove a p âadic analogue of Siegel's theorem, by introducing the p âPythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p âPythagoras number and show that the growth of the p âPythagoras number in finite extensions is bounded
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 -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
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
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
Torsion points and height jumping in higher-dimensional families of abelian varieties
In 1983, Silverman and Tate showed that the set of points in a 1-dimensional family of abelian varieties where a section of infinite order has âsmall heightâ is finite. We conjecture a generalization to higher-dimensional families, where we replace âfiniteâ by ânot Zariski dense.â We show that this conjecture would imply the uniform boundedness conjecture for torsion points on abelian varieties. We then prove a few special cases of this new conjecture.Number theory, Algebra and Geometr