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Let $X$ be a Gorenstein, $ℚ$ -factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of $X$ in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.

The one-motif of an algebraic surface

James A. Carlson (1985)

Compositio Mathematica

The optimality of the Bounded Height Conjecture

Evelina Viada (2009)

Journal de Théorie des Nombres de Bordeaux

In this article we show that the Bounded Height Conjecture is optimal in the sense that, if $V$ is an irreducible subvariety with empty deprived set in a power of an elliptic curve, then every open subset of $V$ does not have bounded height. The Bounded Height Conjecture is known to hold. We also present some examples and remarks.

The osculating hyperquadrics of the three-component distribution in affine space

Marina F. Grebenyuk (2001)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The overconvergence of morphisms of etale $ϕ - Δ$ -spaces on a local field

Nobuo Tsuzuki (1996)

Compositio Mathematica

The $p$ -adic finite Fourier transform and theta functions.

Van Steen, G. (1997)

Bulletin of the Belgian Mathematical Society - Simon Stevin

The $p$ -adic gamma function and the congruences of Atkin and Swinnerton-Dyer

Lucien Van Hamme (1981/1982)

Groupe de travail d'analyse ultramétrique

The $p$ -adic local monodromy theorem for fake annuli

Kiran S. Kedlaya (2007)

Rendiconti del Seminario Matematico della Università di Padova

The $p$ -part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large

Remke Kloosterman (2005)

Journal de Théorie des Nombres de Bordeaux

In this paper we show that for every prime $p \geq 5$ the dimension of the $p$ -torsion in the Tate-Shafarevich group of $E / K$ can be arbitrarily large, where $E$ is an elliptic curve defined over a number field $K$ , with $[K : ℚ]$ bounded by a constant depending only on $p$ . From this we deduce that the dimension of the $p$ -torsion in the Tate-Shafarevich group of $A / ℚ$ can be arbitrarily large, where $A$ is an abelian variety, with $dim A$ bounded by a constant depending only on $p$ .

The $p$ -rank stratification of Artin-Schreier curves

Rachel Pries, Hui June Zhu (2012)

Annales de l’institut Fourier

We study a moduli space ${𝒜𝒮}_{g}$ for Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of characteristic $p$ . We study the stratification of ${𝒜𝒮}_{g}$ by $p$ -rank into strata ${𝒜𝒮}_{g . s}$ of Artin-Schreier curves of genus $g$ with $p$ -rank exactly $s$ . We enumerate the irreducible components of ${𝒜𝒮}_{g, s}$ and find their dimensions. As an application, when $p = 2$ , we prove that every irreducible component of the moduli space of hyperelliptic $k$ -curves with genus $g$ and $2$ -rank $s$ has dimension $g - 1 + s$ . We also determine all pairs $(p, g)$ for...

The period map for hypersurface sections of high degree of an arbitrary variety

Mark L. Green (1985)

Compositio Mathematica

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