Shimizu L-function

In mathematics, the Shimizu L-function, introduced by {{harvs|txt|authorlink=Hideo Shimizu|year=1963|last=Hideo Shimizu}}, is a Dirichlet series associated to a totally real algebraic number field.

{{harvs|txt | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Donnelly | first2=H. | last3=Singer | first3=I. M. | author3-link=Isadore Singer | title=Eta invariants, signature defects of cusps, and values of L-functions | url=https://dx.doi.org/10.2307/2006957 | doi=10.2307/2006957 |mr=707164 | year=1983 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=118 | issue=1 | pages=131–177}}

defined the signature defect of the boundary of a manifold as the eta invariant, the value as s=0 of their eta function, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at s=0 or 1 of a Shimizu L-function.

Definition

Suppose that K is a totally real algebraic number field, M is a lattice in the field, and V is a subgroup of maximal rank of the group of totally positive units preserving the lattice. The Shimizu L-series is given by

: $L(M,V,s) = \sum_{\mu\in \{M-0\}/V} \frac{\operatorname{sign} N(\mu)}{|N(\mu)|^s}$

References

{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Donnelly | first2=H. | last3=Singer | first3=I. M. | title=Geometry and analysis of Shimizu L-functions | jstor=12685 |mr=674920 | year=1982 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=79 | issue=18 | pages=5751 | doi=10.1073/pnas.79.18.5751| pmid=16593231 | pmc=346984 | bibcode=1982PNAS...79.5751A | doi-access=free }}
{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Donnelly | first2=H. | last3=Singer | first3=I. M. | title=Eta invariants, signature defects of cusps, and values of L-functions | doi=10.2307/2006957 |mr=707164 | year=1983 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=118 | issue=1 | pages=131–177| jstor=2006957 }}
{{Citation | last1=Shimizu | first1=Hideo | title=On discontinuous groups operating on the product of the upper half planes | jstor=1970201 |mr=0145106 | year=1963 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=77 | issue=1 | pages=33–71 | doi=10.2307/1970201}}

Category:Zeta and L-functions