Displaying similar documents to “On the arithmetic of the hyperelliptic curve y 2 = x n + a

On arithmetic progressions on Edwards curves

Enrique González-Jiménez (2015)

Acta Arithmetica

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Let m > 0 and a,q ∈ ℚ. Denote by m ( a , q ) the set of rational numbers d such that a, a + q, ..., a + (m-1)q form an arithmetic progression in the Edwards curve E d : x ² + y ² = 1 + d x ² y ² . We study the set m ( a , q ) and we parametrize it by the rational points of an algebraic curve.

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

Nazar Arakelian, Herivelto Borges (2015)

Acta Arithmetica

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For each integer s ≥ 1, we present a family of curves that are q -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be q -Frobenius nonclassical with respect to the linear system of conics. In the q -Frobenius nonclassical cases, we determine the exact number of q -rational points. In the remaining cases, an upper bound for the number of q -rational points will follow from Stöhr-Voloch...

Arithmetic genus of integral space curves

Hao Sun (2018)

Czechoslovak Mathematical Journal

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We give an estimation for the arithmetic genus of an integral space curve which is not contained in a surface of degree k - 1 . Our main technique is the Bogomolov-Gieseker type inequality for 3 proved by Macrì.

An iterative construction for ordinary and very special hyperelliptic curves

Francis J. Sullivan (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si costruiscono famiglie di curve iperellittiche col p —rango della varietà jacobiana uguale a zero. La costruzione sfrutta le proprietà elementari dell’operatore di Cartier e delle estensioni p -cicliche dei corpi con la caratteristica p maggiore di zero.

On covering and quasi-unsplit families of curves

Laurent Bonavero, Cinzia Casagrande, Stéphane Druel (2007)

Journal of the European Mathematical Society

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Given a covering family V of effective 1-cycles on a complex projective variety X , we find conditions allowing one to construct a geometric quotient q : X Y , with q regular on the whole of X , such that every fiber of q is an equivalence class for the equivalence relation naturally defined by V . Among other results, we show that on a normal and -factorial projective variety X with canonical singularities and dim X 4 , every covering and quasi-unsplit family V of rational curves generates a geometric...

Results related to Huppert’s ρ - σ conjecture

Xia Xu, Yong Yang (2023)

Czechoslovak Mathematical Journal

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We improve a few results related to Huppert’s ρ - σ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups.

On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman, Rebecca Lehman, Maxim Leyenson, Mina Teicher (2012)

Journal of the European Mathematical Society

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The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in 3 . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, B and E , we give a necessary and sufficient condition for B to be the branch curve of a surface X in N and E to be the image of the double curve of a 3 -model of X . In the classical Segre theory, a...

Explicit Teichmüller curves with complementary series

Carlos Matheus, Gabriela Weitze-Schmithüsen (2013)

Bulletin de la Société Mathématique de France

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We construct an explicit family of arithmetic Teichmüller curves 𝒞 2 k , k , supporting SL ( 2 , ) -invariant probabilities μ 2 k such that the associated SL ( 2 , ) -representation on  L 2 ( 𝒞 2 k , μ 2 k ) has complementary series for every k 3 . Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves 𝒞 2 k has arbitrarily slow rate of exponential mixing.

Some examples of 5 and 7 descent for elliptic curves over Q

Tom Fisher (2001)

Journal of the European Mathematical Society

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We perform descent calculations for the families of elliptic curves over Q with a rational point of order n = 5 or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over Q may become arbitrarily large.

An iterative construction for ordinary and very special hyperelliptic curves

Francis J. Sullivan (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Si costruiscono famiglie di curve iperellittiche col p —rango della varietà jacobiana uguale a zero. La costruzione sfrutta le proprietà elementari dell’operatore di Cartier e delle estensioni p -cicliche dei corpi con la caratteristica p maggiore di zero.

Numerical characterization of nef arithmetic divisors on arithmetic surfaces

Atsushi Moriwaki (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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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 ¯ ) .