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Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion . In the first part, we apply ideas from the proof of by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if are such that , then . This allows us to conclude, among other things, that and .
Si un système d’équations polynomiales à coefficients entiers admet une solution dans , il en admet sur tout complété -adique ou réel de . La réciproque a été démontrée par Hasse pour les quadriques, mais elle est fausse en général. Une grande partie des contre-exemples connus peuvent être expliqués à l’aide de l’obstruction de Brauer-Manin, basée sur la théorie du corps de classe. Il est donc naturel de se demander si, pour certaines classes de variétés, cette obstruction est la seule. Le but...
Watkins has conjectured that if is the rank of the group of rational points of an elliptic curve over the rationals, then divides the modular parametrisation degree. We show, for a certain class of , chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain -adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others, to prove...
Let be a morphism of a variety defined over a number field , let be a -subvariety, and let be the orbit of a point . We describe a local-global principle for the intersection . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of are Brauer–Manin unobstructed for power maps on in two cases: (1) is a translate of a torus. (2) is a line and has a preperiodic coordinate. A key tool in the proofs is the classical...
Let be a family of elliptic curves over , where is a positive integer and , are distinct odd primes. We study the torsion part and the rank of . More specifically, we prove that the torsion subgroup of is trivial and the -rank of this family is at least 2, whenever , and with neither nor dividing .
A well known theorem of Mestre and Schoof implies that the order of an elliptic curve over a prime field can be uniquely determined by computing the orders of a few points on and its quadratic twist, provided that . We extend this result to all finite fields with , and all prime fields with .
This paper studies a two-variable zeta function attached to an algebraic
number field , introduced by van der Geer and Schoof, which is based on an analogue of
the Riemann-Roch theorem for number fields using Arakelov divisors. When this
function becomes the completed Dedekind zeta function of the field .
The function is a meromorphic function of two complex variables with polar divisor , and it satisfies the functional equation . We consider the
special case , where for this function...
Let and a,q ∈ ℚ. Denote by the set of rational numbers d such that a, a + q, ..., a + (m-1)q form an arithmetic progression in the Edwards curve . We study the set and we parametrize it by the rational points of an algebraic curve.
It is well known that duality theorems are of utmost importance for the arithmetic of local and global fields and that Brauer groups appear in this context unavoidably. The key word here is class field theory.In this paper we want to make evident that these topics play an important role in public key cryptopgraphy, too. Here the key words are Discrete Logarithm systems with bilinear structures.Almost all public key crypto systems used today based on discrete logarithms use the ideal class groups...
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