Some Remarks on Invariants of Abelian Varieties Concerning the Reduction Theory

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Throughout this paper, we are concerned about an abelian variety A defined over a field K complete under a (non-trivial) discrete valuation whose residue field k is algebraically closed. In §1., basing on Neron's work [5], we estimate the cardinality of K-rational l-divisible points of the origin O on A (i.e. the cardinality of the set A_l(K)={x∈A|x is rational over K and satisfies the condition l・x=O}), where l is a prime number different from the characteristic of k (§1, Proposition 1). Moreover we talk about the vanishing-ness of the cohomology group H^n(K, A) (n>0) of G_K with coefficients in A(K_s) in the sense of Lang-Tate [3] or Serre [8], where K_s denotes the separable algebraic closure of K and G_K denotes the galois group with Krull-topology for the galois extension K_s/K and A(K_s) denotes the G_K-module with discrete topology consisting of all K_s rational points on A (§1, Proposition 2). In §2, when dim A=1, we survey values of some invariants under all isomorphisms defined over K (called K-isomorphisms) of the abelian variety A and the situation of its reduction over K by means of Neron's p-standard K-model of A, and we arrange them as a list.
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