The Néron component series of an abelian variety

Abstract

We introduce the Néron component series of an abelian variety A over a complete discretely valued field. This is a power series in Z[[T ]], which measures the behaviour of the number of components of the Néron model of A under tame ramification of the base field. If A is tamely ramified, then we prove that the Néron component series is rational. It has a pole at T = 1, whose order equals one plus the potential toric rank of A. This result is a crucial ingredient of our proof of the motivic monodromy conjecture for abelian varieties. We expect that it extends to the wildly ramified case; we prove this if A is an elliptic curve, and if A has potential purely multiplicative reduction.

Néron Models and Base Change

“…Then C is tamely ramified. By Proposition 2.2.4, the integer e(C) equals the degree of the minimal extension of K over which C acquires semi-stable reduction, so that the result follows from [HN10,5.7]. …”

“…Since each of the connected components of A (d) k is isomorphic to the identity component A (d) that we introduced in [HN10] (there it was denoted S φ (A; T )). This series measures how the number of Néron components varies under tame extensions of K. We were able to prove in [HN10,6.5] that it is a rational function when A is tamely ramified or A has potential multiplicative reduction.…”

“…This series measures how the number of Néron components varies under tame extensions of K. We were able to prove in [HN10,6.5] that it is a rational function when A is tamely ramified or A has potential multiplicative reduction. This was a key ingredient of our proof that the motivic zeta function Z A (T ) is rational if A is tamely ramified [HN11a].…”

“…The dimension of G tor is called the reductive rank of G and denoted by ρ(G). By [HN10,3.4], the reductive rank of G/G tor is zero. The algebraic group G is a semi-abelian F -variety if and only if G/G tor is an abelian variety; in that case, we denote this quotient by G ab .…”

“…(2.1.2) Every F -torus T has a unique maximal split subtorus T spl [HN10,3.5]. The cocharacter module of T spl is the submodule of Galois-invariant elements of the cocharacter module of T .…”