We describe certain birational modifications of nodal relative curves over a regular base of arbitrary dimension. Using these modifications, we provide criteria for the existence of Néron models of... Show moreWe describe certain birational modifications of nodal relative curves over a regular base of arbitrary dimension. Using these modifications, we provide criteria for the existence of Néron models of nodal curves and their Jacobians over a regular base of arbitrary dimension, and describe the Néron models explicitly when they exist. Then, we study the descent behavior of Néron models of abelian varieties under tamely ramified and surjective maps between regular schemes. Show less
Chapter 1,contains the numerical verification of the Birch and Swinnerton-Dyer conjecture for hundreds of Jacobians of hyperelliptic curves of genus 2, 3, 4 and 5. Chapter 2 treats the... Show moreChapter 1,contains the numerical verification of the Birch and Swinnerton-Dyer conjecture for hundreds of Jacobians of hyperelliptic curves of genus 2, 3, 4 and 5. Chapter 2 treats the equivalence of BSD for a certain elliptic curve over Q(∜5), and a pair of hyperelliptic curves over Q. In chapter 3, ordinary Galois covers of smooth curves are constructed from ordinary Galois covers of semi-stable curves. Finally, in chapter 4, the statistics of ordinary reduction for hyperelliptic curves is considered. Show less
In the first part, we introduce a new condition, called toric-additivity, on a family of abelian varieties degenerating to a semi-abelian scheme over a normal crossing divisor. The condition... Show moreIn the first part, we introduce a new condition, called toric-additivity, on a family of abelian varieties degenerating to a semi-abelian scheme over a normal crossing divisor. The condition depends only on the l-adic Tate module of the generic fibre, for a prime l invertible on S. We show that toric-additivity is strictly related to the property of existence of Neron models. In the second part, we consider the case of a family of nodal curves over a discrete valuation ring, having split singularities. We say that such a family is semi-factorial if every line bundle on the generic bre extends to a line bundle on the total space. We give a necessary and sufficient condition for semifactoriality, in terms of combinatorics of the dual graph of the special fibre. Show less
