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. © 2010 Springer-Verlag.
Halle L.H., Nicaise J. (2010). The Néron component series of an abelian variety. MATHEMATISCHE ANNALEN, 348(3), 749-778 [10.1007/s00208-010-0495-5].
The Néron component series of an abelian variety
Halle L. H.;
2010
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. © 2010 Springer-Verlag.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
