Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Local-global Summation Formula for Abelian Varieties   Let $K$ be a field finitely generated over ${\Q}$, and $A$ an Abelian variety
defined over $K$. Then by the Mordell-Weil Theorem, the set of rational points
$A(K)$ is a finitely-generated Abelian group. In this paper, assuming Tate's
Conjecture on algebraic cycles, we prove a limit formula for the Mordell-Weil
rank of an arbitrary family of Abelian varieties $A$ over a number field $k$;
this is the Abelian fibration analogue of the Nagao formula for elliptic
surfaces $E$, originally conjectured by Nagao, and proven by Rosen and
Silverman to be equivalent to Tate's Conjecture for $E$. We also give a short
exact sequence relating the Picard Varieties of the family $A$, the parameter
space, and the generic fiber, and use this to obtain an isomorphism (modulo
torsion) relating the Neron-Severi group of $A$ to the Mordell-Weil group of
$A$.

32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th