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Covers in p -adic analytic geometry and log covers I: Cospecialization of the ( p ′ ) -tempered fundamental group for a family of curves

Emmanuel Lepage (2013)

Annales de l’institut Fourier

The tempered fundamental group of a p -adic analytic space classifies covers that are dominated by a topological cover (for the Berkovich topology) of a finite étale cover of the space. Here we construct cospecialization homomorphisms between ( p ′ ) versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log reduction and...

Critical and ramification points of the modular parametrization of an elliptic curve

Christophe Delaunay (2005)

Journal de Théorie des Nombres de Bordeaux

Let E be an elliptic curve defined over ℚ with conductor N and denote by ϕ the modular parametrization: ϕ : X 0 ( N ) → E ( ℂ ) . In this paper, we are concerned with the critical and ramification points of ϕ . In particular, we explain how we can obtain a more or less experimental study of these points.

Cubic forms, powers of primes and the Kraus method

Andrzej Dąbrowski, Tomasz Jędrzejak, Karolina Krawciów (2012)

Colloquium Mathematicae

We consider the Diophantine equation ( x + y ) ( x ² + B x y + y ² ) = D z p , where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).

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