Algorithmic methods for arithmetic surfaces and regular, minimal models

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Regular and minimal models of algebraic curves over number fields are arithmetic surfaces that play an important role in arithmetic geometry. This research project aims at developing algorithms for such arithmetic surfaces and for the computation of regular and minimal models. The main topics are a desingularisation procedure following Lipman, functionality for the intersection pairing, exceptional divisors, blow ups and blow downs. On the basis of these algorithms applications to the Birch and Swinnerton-Dyer conjecture and other related areas are finally investigated.