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The circle method and pairs of quadratic forms

Henryk Iwaniec, Ritabrata Munshi (2010)

Journal de Théorie des Nombres de Bordeaux

We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.

The intersection of a curve with algebraic subgroups in a product of elliptic curves

Evelina Viada (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider an irreducible curve 𝒞 in E n , where E is an elliptic curve and 𝒞 and E are both defined over ¯ . Assuming that 𝒞 is not contained in any translate of a proper algebraic subgroup of E n , we show that the points of the union 𝒞 A ( ¯ ) , where A ranges over all proper algebraic subgroups of E n , form a set of bounded canonical height. Furthermore, if E has Complex Multiplication then the set 𝒞 A ( ¯ ) , for A ranging over all algebraic subgroups of E n of codimension at least 2 , is finite. If E has no Complex Multiplication...

Trivial points on towers of curves

Xavier Xarles (2013)

Journal de Théorie des Nombres de Bordeaux

In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.

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