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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 with conductor N and denote by ϕ the modular parametrization: ϕ : X 0 ( N ) E ( ) . 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 differential forms and the group law on the Jacobian of a real algebraic curve

J. Huisman (2003)

Bollettino dell'Unione Matematica Italiana

In an earlier paper [6], we gave an explicit geometric description of the group law on the neutral component of the set of real points of the Jacobian of a smooth quartic curve. Here, we generalize this description to curves of higher genus. We get a description of the group law on the neutral component of the set of real points of the Jacobian of a smooth curve in terms of cubic differential forms. When applied to canonical curves, one gets an explicit geometric description of this group law by...

Cubic surfaces with a Galois invariant double-six

Andreas-Stephan Elsenhans, Jörg Jahnel (2010)

Open Mathematics

We present a method to construct non-singular cubic surfaces over ℚ with a Galois invariant double-six. We start with cubic surfaces in the hexahedral form of L. Cremona and Th. Reye. For these, we develop an explicit version of Galois descent.

Curves in P2(C) with 1-dimensional symmetry.

A. A. du Plessis, Charles Terence Clegg Wall (1999)

Revista Matemática Complutense

In a previous paper we showed that the existence of a 1-parameter symmetry group of a hypersurface X in projective space was equivalent to failure of versality of a certain unfolding. Here we study in detail (reduced) plane curves of degree d ≥ 3, excluding the trivial case of cones. We enumerate all possible group actions -these have to be either semisimple or unipotent- for any degree d. A 2-parameter group can only occur if d = 3. Explicit lists of singularities of the corresponding curves are...

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