In
mathematics, the Shimizu ''L''-function, introduced by , is a
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analyti ...
associated to a
totally real
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyn ...
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
.
defined the
signature defect In mathematics, 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 surfaces.
defined the signature d ...
of the boundary of a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
as the
eta invariant In mathematics, the eta invariant of a self-adjoint elliptic differential operator on a compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbers are often infinite so are d ...
, 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
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 variet ...
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
:
References
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Zeta and L-functions
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