The Néron component series of an abelian variety
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The Néron component series of an abelian variety. / Halle, Lars Halvard; Nicaise, Johannes.
I: Mathematische Annalen, Bind 348, Nr. 3, 08.03.2010, s. 749-778.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - The Néron component series of an abelian variety
AU - Halle, Lars Halvard
AU - Nicaise, Johannes
PY - 2010/3/8
Y1 - 2010/3/8
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=77955918179&partnerID=8YFLogxK
U2 - 10.1007/s00208-010-0495-5
DO - 10.1007/s00208-010-0495-5
M3 - Journal article
AN - SCOPUS:77955918179
VL - 348
SP - 749
EP - 778
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 3
ER -
ID: 233909918