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In this paper we consider questions of the following type. Let $k$ be a base field and $K / k$ be a field extension. Given a geometric object $X$ over a field $K$ (e.g. a smooth curve of genus $g$ ), what is the least transcendence degree of a field of definition of $X$ over the base field $k$ ? In other words, how many independent parameters are needed to define $X$ ? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete answer to...

Eště o niektorých ďalších vlastnostiach kriviek triedy P a PP

Jozef Oboňa, Nikolaj Podtjagin (1973)

Časopis pro pěstování matematiky

Estimating isogenies on elliptic curves.

D.W. Masser, G. Wüstholz (1990)

Inventiones mathematicae

Etale cohomology of henselian rings and cohomology of abstract Riemann surfaces of fields.

Roland Huber (1993)

Mathematische Annalen

Etale coverings of a Mumford curve

Marius Van Der Put (1983)

Annales de l'institut Fourier

Let the field $K$ be complete w.r.t. a non-archimedean valuation. Let $X / K$ be a Mumford curve, i.e. the irreducible components of the stable reduction of $X$ have genus 0. The abelian etale coverings of $X$ are constructed using the analytic uniformization $Ω \to X$ and the theta-functions on $X$ . For a local field $K$ one rediscovers $G$ . Frey’s description of the maximal abelian unramified extension of the field of rational functions of $X$ .

Étale Galois covers of affine smooth cuves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture.

Florian Pop (1995)

Inventiones mathematicae

Étude de la courbe elliptique $y^{2} = 4 x^{3} - 27 {((3 + \sqrt{- 19}) / 2)}^{2}$ et monogénéité de certains anneaux d’entiers

Vincent Fleckinger (1989)

Journal de théorie des nombres de Bordeaux

Étude des intégrales abéliennes de troisième espèce

D. Emmanuel (1879)

Annales scientifiques de l'École Normale Supérieure

Étude sur les relations algébriques entre les fonctions hyperelliptiques de genre 3

Brunel (1883)

Annales scientifiques de l'École Normale Supérieure

Euclidean algorithm and Kummer covers with many points.

Garzón, Alvaro (2003)

Revista Colombiana de Matemáticas

Euler characteristics of moduli spaces of curves

Gilberto Bini, John Harer (2011)

Journal of the European Mathematical Society

Let $M_{g}^{n}$ be the moduli space of $n$ -pointed Riemann surfaces of genus $g$ . Denote by $\bar{M_{g}^{n}}$ the Deligne-Mumford compactification of $M_{g}^{n}$ . In the present paper, we calculate the orbifold and the ordinary Euler characteristic of $\bar{M_{g}^{n}}$ for any $g$ and $n$ such that $n > 2 - 2 g$ .

Euler's concordant forms

Ken Ono (1996)

Acta Arithmetica

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