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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 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 Mathé ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 variety is an algebraic variety obtained by taking a quotient of a product of multiple copies of the upper half-plane by a Hilbert modular group. Hilbert modular surfaces were first described by using some unpublished notes written by David Hilbert about 10 years before. Definitions If ''R'' is the ring of integers of a real quadratic field, then the Hilbert modular group SL2(''R'') acts on the product ''H''×''H'' of two copies of the upper half plane ''H''. There are several birationally equivalent surfaces related to this action, any of which may be called Hilbert modular surfaces: *The surface ''X'' is the quotient of ''H''×''H'' by SL2(''R''); it is not compact and usually has quotient singularities coming from point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 defined using zeta function regularization. It was introduced by who used it to extend the Hirzebruch signature theorem to manifolds with boundary. The name comes from the fact that it is a generalization of the Dirichlet eta function. They also later used the eta invariant of a self-adjoint operator to define the eta invariant of a compact odd-dimensional smooth manifold. defined the signature defect of the boundary of a manifold as the eta invariant, 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 The eta invariant of self-adjoint operator ''A'' is given by ''η''''A''(0), where ''η'' is the an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shimizu L-function
In mathematics, the Shimizu ''L''-function, introduced by , is a Dirichlet series associated to a totally real algebraic number field. 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 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 ... 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 :L(M,V,s) = \sum_ \frac References * * * Zeta and L-functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
