nLab Shimizu L-function

Idea

(…)

Definition

Given a totally real algebraic number field $K$, a lattice $M$ on it and a subgroup $V$ of maximal rank of the subgroup of totally positive units preserving the lattice, its Shimizu L-function is given by:

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

Properties

Michael Atiyah, Harold Donnelly and Isadore Singer defined the signature defect of the boundary of a manifold as the eta invariant, the value at $s=0$ of their eta function, and used this to show that Hirzebruch’s signature defect of a cusp of a Hilbert modular variety can be expressed in terms of the value at $s=0$ or $s=1$ of a Shimizu L-function.

Related concepts

• L-function

References

The notion is due to:

• Hideo Shimizu, On discontinuous groups operating on the product of the upper half planes, Annals of Mathematics. 77 1 (1963) 5751 [ISSN 0003-486X, doi:10.2307/1970201, PMID 16593231, PMC 346984 , bibcode:1982PNAS…79.5751A, jstor:1970201]

Further developments:

• Michael Atiyah, Harold Donnelly, Isadore Singer: Geometry and analysis of Shimizu L-functions, Proceedings of the National Academy of Sciences of the United States of America. 79 18 (1982) 5751 [ISSN 0027-8424, doi:10.1073/pnas.79.18.5751, PMID 16593231, PMC 346984, bibcode:1982PNAS…79.5751A, jstor:12685]

• Michael Atiyah, Harold Donnelly, Isadore Singer: Eta invariants, signature defects of cusps, and values of L-functions, Annals of Mathematics. 118 1 (1982) 131–177 [ISSN 0003-486X, doi:10.2307/2006957, jstor:2006957]

See also:

• Wikipedia, Shimizu L-function