Shimizu L-function

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In mathematics, the Shimizu L-function, introduced by Hideo Shimizu in 1963,^[1] is a Dirichlet series associated to a totally real algebraic number field. Michael Francis Atiyah, H. Donnelly, and I. M. Singer (1983)^[2] 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

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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

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^ Shimizu, Hideo (January 1963). "On Discontinuous Groups Operating on the Product of the Upper Half Planes". The Annals of Mathematics. 77 (1): 33. doi:10.2307/1970201.
^ Atiyah, Michael Francis; Donnelly, H.; Singer, I. M. (1983), "Eta invariants, signature defects of cusps, and values of L-functions", Annals of Mathematics, Second Series, 118 (1): 131-177, doi:10.2307/2006957, ISSN 0003-486X, JSTOR 2006957, MR 0707164

Definition

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