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The rank of hyperelliptic Jacobians in families of quadratic twists

Sebastian Petersen (2006)

Journal de Théorie des Nombres de Bordeaux

The variation of the rank of elliptic curves over ℚ in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over ℚ admits infinitely many quadratic twists of rank ≥ 1 . Most elliptic curves have even infinitely many twists of rank ≥ 2 and examples of elliptic curves with infinitely many twists of rank ≥ 4 are known. There are also certain density results. This paper studies the variation of the...

The real field with the rational points of an elliptic curve

Ayhan Günaydın, Philipp Hieronymi (2011)

Fundamenta Mathematicae

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.

The reciprocal of the beta function and G L ( n , ℝ ) Whittaker functions

Eric Stade (1994)

Annales de l'institut Fourier

In this paper we derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler’s beta function. This expression is similar in form to several well-known integrals for the beta function itself.We then apply our new formula to the study of G L ( n , ℝ ) Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral representations of “fundamental”...

The reciprocal super Catalan matrix

Helmut Prodinger (2015)

Special Matrices

The reciprocal super Catalan matrix has entries [...] . Explicit formulæ for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, q-analogues are also presented.

The reduced ideals of a special order in a pure cubic number field

Abdelmalek Azizi, Jamal Benamara, Moulay Chrif Ismaili, Mohammed Talbi (2020)

Archivum Mathematicum

Let K = ℚ ( θ ) be a pure cubic field, with θ 3 = D , where D is a cube-free integer. We will determine the reduced ideals of the order 𝒪 = ℤ [ θ ] of K which coincides with the maximal order of K in the case where D is square-free and ¬ ≡ ± 1 ( mod 9 ) .

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