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S -integral points on elliptic curves - Notes on a paper of B. M. M. de Weger

Emanuel Herrmann, Attila Pethö (2001)

Journal de théorie des nombres de Bordeaux

In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and p -adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.

S -integral solutions to a Weierstrass equation

Benjamin M. M. de Weger (1997)

Journal de théorie des nombres de Bordeaux

The rational solutions with as denominators powers of 2 to the elliptic diophantine equation y 2 = x 3 - 228 x + 848 are determined. An idea of Yuri Bilu is applied, which avoids Thue and Thue-Mahler equations, and deduces four-term ( S -) unit equations with special properties, that are solved by linear forms in real and p -adic logarithms.

Selmer groups for elliptic curves in l d -extensions of function fields of characteristic p

Andrea Bandini, Ignazio Longhi (2009)

Annales de l’institut Fourier

Let F be a function field of characteristic p > 0 , / F a l d -extension (for some prime l p ) and E / F a non-isotrivial elliptic curve. We study the behaviour of the r -parts of the Selmer groups ( r any prime) in the subextensions of via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of / F .

Semistable reduction and torsion subgroups of abelian varieties

Alice Silverberg, Yuri G. Zarhin (1995)

Annales de l'institut Fourier

The main result of this paper implies that if an abelian variety over a field F has a maximal isotropic subgroup of n -torsion points all of which are defined over F , and n 5 , then the abelian variety has semistable reduction away from n . This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its n -torsion points are defined over a field F and n 3 , then the abelian variety has semistable reduction away from n . We also give information about the Néron models...

Set of Points on Elliptic Curve in Projective Coordinates

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2011)

Formalized Mathematics

In this article, we formalize a set of points on an elliptic curve over GF(p). Elliptic curve cryptography [10], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.

Shimura varieties with Γ 1 ( p ) -level via Hecke algebra isomorphisms: the Drinfeld case

Thomas J. Haines, Michael Rapoport (2012)

Annales scientifiques de l'École Normale Supérieure

We study the local factor at  p of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at  p by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining their images under the Hecke algebra isomorphisms of Goldstein, Morris, and Roche.

Siegel’s theorem and the Shafarevich conjecture

Aaron Levin (2012)

Journal de Théorie des Nombres de Bordeaux

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k , one can effectively compute the set of isomorphism classes of hyperelliptic curves over k with good reduction outside S . We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus g would imply an effective version of Siegel’s theorem for integral points on...

Sieve methods for varieties over finite fields and arithmetic schemes

Bjorn Poonen (2007)

Journal de Théorie des Nombres de Bordeaux

Classical sieve methods of analytic number theory have recently been adapted to a geometric setting. In the new setting, the primes are replaced by the closed points of a variety over a finite field or more generally of a scheme of finite type over . We will present the method and some of the surprising results that have been proved using it. For instance, the probability that a plane curve over 𝔽 2 is smooth is asymptotically 21 / 64 as its degree tends to infinity. Much of this paper is an exposition...

Sign functions of imaginary quadratic fields and applications

Hassan Oukhaba (2005)

Annales de l’institut Fourier

We propose a definition of sign of imaginary quadratic fields. We give an example of such functions, and use it to define new invariants that are roots of the classical Ramachandra invariants. Also we introduce signed ordinary distributions and compute their signed cohomology by using Anderson's theory of double complex.

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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