ncku-talk2007-05-31

NCKU Math colloquium

DATE 2007-05-31　16:10-17:00

PLACE 數學館地下室演講廳

SPEAKER Liang-Chung Hsia（）

TITLE A quotient of elliptic curves - weak Neron models for Lattes maps

ABSTRACT Let R be a discrete valuation ring and let K be its field of fractions. Let \phi : V \to V be a morphism on a smooth variety V over K. In studying local canonical height associated to \phi, G. Call and J. Silverman generalized Neron's theory by introducing the notion of a weak Neron model over R for the pair(V/K,\phi). If V is not an abelian variety, then a weak Neron model for (V/K,\phi) may not exist in general.
In this talk, we consider the case where V= P^1 and \phi is a morphism on P^1 over K. We will discuss possible obstructions to the existence of a weak Neron model for (P^1/K,\phi). On the other hand, we will look at an important family, called the Lattes maps which come from quotient of pairs (E, \psi) where E is an elliptic curve over K and \psi is an endomorphism of E. In a joint work with R. Benedetto, we show that for Lattes maps, weak Neron models always exist.
We will also discuss the dynamics of the Lattes maps on the Berkovich space P_{Berk}^1 associated to the projective line.

NCKU Math colloquium

DATE	2007-05-31　16:10-17:00

PLACE	數學館地下室演講廳

SPEAKER	Liang-Chung Hsia（）

TITLE	A quotient of elliptic curves - weak Neron models for Lattes maps

ABSTRACT	Let R be a discrete valuation ring and let K be its field of fractions. Let \phi : V \to V be a morphism on a smooth variety V over K. In studying local canonical height associated to \phi, G. Call and J. Silverman generalized Neron's theory by introducing the notion of a weak Neron model over R for the pair(V/K,\phi). If V is not an abelian variety, then a weak Neron model for (V/K,\phi) may not exist in general. In this talk, we consider the case where V= P^1 and \phi is a morphism on P^1 over K. We will discuss possible obstructions to the existence of a weak Neron model for (P^1/K,\phi). On the other hand, we will look at an important family, called the Lattes maps which come from quotient of pairs (E, \psi) where E is an elliptic curve over K and \psi is an endomorphism of E. In a joint work with R. Benedetto, we show that for Lattes maps, weak Neron models always exist. We will also discuss the dynamics of the Lattes maps on the Berkovich space P_{Berk}^1 associated to the projective line.