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Numerical characterization of nef arithmetic divisors on arithmetic surfaces

Atsushi Moriwaki (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper, we give a numerical characterization of nef arithmetic -Cartier divisors of C 0 -type on an arithmetic surface. Namely an arithmetic -Cartier divisor D ¯ of C 0 -type is nef if and only if D ¯ is pseudo-effective and deg ^ ( D ¯ 2 ) = vol ^ ( D ¯ ) .

Obstruction sets and extensions of groups

Francesca Balestrieri (2016)

Acta Arithmetica

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 X ( k ) é t , B r X ( k ) B r . In the first part, we apply ideas from the proof of X ( k ) é t , B r = X ( k ) k by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if k are such that E x t ( , k ) , then X ( k ) = X ( k ) . This allows us to conclude, among other things, that X ( k ) é t , B r = X ( k ) k and X ( k ) S o l , B r = X ( k ) S o l k .

Obstructions au principe de Hasse et à l’approximation faible

Emmanuel Peyre (2003/2004)

Séminaire Bourbaki

Si un système d’équations polynomiales à coefficients entiers admet une solution dans 𝐐 n , il en admet sur tout complété p -adique ou réel de 𝐐 . 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...

On a conjecture of Watkins

Neil Dummigan (2006)

Journal de Théorie des Nombres de Bordeaux

Watkins has conjectured that if R is the rank of the group of rational points of an elliptic curve E over the rationals, then 2 R divides the modular parametrisation degree. We show, for a certain class of E , chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain 2 -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...

On a dynamical Brauer–Manin obstruction

Liang-Chung Hsia, Joseph Silverman (2009)

Journal de Théorie des Nombres de Bordeaux

Let ϕ : X X be a morphism of a variety defined over a number field  K , let  V X be a K -subvariety, and let  𝒪 ϕ ( P ) = { ϕ n ( P ) : n 0 } be the orbit of a point  P X ( K ) . We describe a local-global principle for the intersection  V 𝒪 ϕ ( P ) . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of  V ( K ) are Brauer–Manin unobstructed for power maps on  2 in two cases: (1)  V is a translate of a torus. (2)  V is a line and  P has a preperiodic coordinate. A key tool in the proofs is the classical...

On a family of elliptic curves of rank at least 2

Kalyan Chakraborty, Richa Sharma (2022)

Czechoslovak Mathematical Journal

Let C m : y 2 = x 3 - m 2 x + p 2 q 2 be a family of elliptic curves over , where m is a positive integer and p , q are distinct odd primes. We study the torsion part and the rank of C m ( ) . More specifically, we prove that the torsion subgroup of C m ( ) is trivial and the -rank of this family is at least 2, whenever m ¬ 0 ( mod 3 ) , m ¬ 0 ( mod 4 ) and m 2 ( mod 64 ) with neither p nor q dividing m .

On a theorem of Mestre and Schoof

John E. Cremona, Andrew V. Sutherland (2010)

Journal de Théorie des Nombres de Bordeaux

A well known theorem of Mestre and Schoof implies that the order of an elliptic curve E over a prime field 𝔽 q can be uniquely determined by computing the orders of a few points on E and its quadratic twist, provided that q > 229 . We extend this result to all finite fields with q > 49 , and all prime fields with q > 29 .

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