nLab Shimizu L-function

Redirected from "Shimizu L-functions".
Contents

Contents

Idea

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Definition

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

L(M,V,s)= μ{M0}/VsignN(μ)|N(μ)| s. 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=0s=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=0s=0 or s=1s=1 of a Shimizu 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:

See also:

Last revised on May 8, 2024 at 14:43:46. See the history of this page for a list of all contributions to it.