TY - JOUR

T1 - On the syntomic regulator for K1 of a surface

AU - Besser, Amnon

N1 - Funding Information:
Acknowledgements. Thanks go first and foremost to A. Langer, whose questions relating to his work sparked our interest in this problem, for his continuous interest in this work. We thank N. Katz for some useful discussions. We thank the anonymous referee for some helpful comments. We would also like to thank the Institute for Advanced Study in Princeton where most of this work was done, and the Bell Companies Fellowship and the James D. Wolfensohn fund for financial support while at the Institute.

PY - 2012/8/1

Y1 - 2012/8/1

N2 - We consider elements of K1(S), where S is a proper surface over a p-adic field with good reduction, which are given by a formal sum Σ(Zi, fi) with Zi curves in S and fi rational functions on the Zi in such a way that the sum of the divisors of the fi is 0 on S. Assuming compatibility of pushforwards in syntomic and motivic cohomologies, our result computes the syntomic regulator of such an element, interpreted as a functional on HdR2(S), when evaluated on the cup product ω∪[η] of a holomorphic form ω by the first cohomology class of a form of the second kind η. The result is Σi〈Fη, log(fi); Fω〉gl,Zi, where Fω and Fη are Coleman integrals of ω and η, respectively, and the symbol in brackets is the global triple index, as defined in our previous work.

AB - We consider elements of K1(S), where S is a proper surface over a p-adic field with good reduction, which are given by a formal sum Σ(Zi, fi) with Zi curves in S and fi rational functions on the Zi in such a way that the sum of the divisors of the fi is 0 on S. Assuming compatibility of pushforwards in syntomic and motivic cohomologies, our result computes the syntomic regulator of such an element, interpreted as a functional on HdR2(S), when evaluated on the cup product ω∪[η] of a holomorphic form ω by the first cohomology class of a form of the second kind η. The result is Σi〈Fη, log(fi); Fω〉gl,Zi, where Fω and Fη are Coleman integrals of ω and η, respectively, and the symbol in brackets is the global triple index, as defined in our previous work.

UR - http://www.scopus.com/inward/record.url?scp=84865546974&partnerID=8YFLogxK

U2 - 10.1007/s11856-011-0188-0

DO - 10.1007/s11856-011-0188-0

M3 - Article

AN - SCOPUS:84865546974

SN - 0021-2172

VL - 190

SP - 29

EP - 66

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -