On twisted zeta-functions at s=0 and partial zeta-functions at s=1

TY - JOUR

T1 - On twisted zeta-functions at s=0 and partial zeta-functions at s=1

AU - Solomon, D

PY - 2008/1

Y1 - 2008/1

N2 - Let K be an abelian extension of a totally real number field k, K+ its maximal real subfield and G=Gal(K/k). We have previously used twisted zeta-functions to define a meromorphic -valued function ΦK/k(s) in a way similar to the use of partial zeta-functions to define the better-known function ΘK/k(s). For each prime number p, we now show how the value ΦK/k(0) combines with a p-adic regulator of semilocal units to define a natural -submodule of which we denote . If p is odd and splits in k, our main theorem states that is (at least) contained in . Thanks to a precise relation between ΦK/k(1−s) and ΘK/k(s), this theorem can be reformulated in terms of (the minus part of) ΘK/k(s) at s=1, making it an analogue of Deligne–Ribet and Cassou-Noguès' well-known integrality result concerning ΘK/k(0). We also formulate some conjectures including a congruence involving Hilbert symbols that links with the Rubin–Stark Conjecture for K+/k.

AB - Let K be an abelian extension of a totally real number field k, K+ its maximal real subfield and G=Gal(K/k). We have previously used twisted zeta-functions to define a meromorphic -valued function ΦK/k(s) in a way similar to the use of partial zeta-functions to define the better-known function ΘK/k(s). For each prime number p, we now show how the value ΦK/k(0) combines with a p-adic regulator of semilocal units to define a natural -submodule of which we denote . If p is odd and splits in k, our main theorem states that is (at least) contained in . Thanks to a precise relation between ΦK/k(1−s) and ΘK/k(s), this theorem can be reformulated in terms of (the minus part of) ΘK/k(s) at s=1, making it an analogue of Deligne–Ribet and Cassou-Noguès' well-known integrality result concerning ΘK/k(0). We also formulate some conjectures including a congruence involving Hilbert symbols that links with the Rubin–Stark Conjecture for K+/k.

U2 - 10.1016/j.jnt.2007.03.010

DO - 10.1016/j.jnt.2007.03.010

M3 - Article

VL - 128

SP - 105

EP - 143

JO - Journal of Number Theory

JF - Journal of Number Theory

IS - 1

ER -