mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the signature defect of a singularity measures the correction that a singularity contributes to the signature theorem. introduced the signature defect for the cusp singularities of

Hilbert modular surface In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular vari ...

s. 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.

References

* *{{Citation , last1=Hirzebruch , first1=Friedrich E. P. , title=Hilbert modular surfaces , doi=10.5169/seals-46292 , mr=0393045 , year=1973 , journal=L'Enseignement Mathématique , series=2e Série , issn=0013-8584 , volume=19 , pages=183–281 Singularity theory