In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.
Siegel modular forms are holomorphic functions on the set of symmetric n × n matrices with positive definite imaginary part; the forms must satisfy an automorphy condition. Siegel modular forms can be thought of as multivariable modular forms, i.e. as special functions of several complex variables.
Siegel modular forms were first investigated by Carl Ludwig Siegel (1939) for the purpose of studying quadratic forms analytically. These primarily arise in various branches of number theory, such as arithmetic geometry and elliptic cohomology. Siegel modular forms have also been used in some areas of physics, such as conformal field theory and black hole thermodynamics in string theory.
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In mathematics, Siegelmodularforms are a major type of automorphic form. These generalize conventional elliptic modularforms which are closely related...
In mathematics, a modularform is a (complex) analytic function on the upper half-plane, H {\displaystyle \,{\mathcal {H}}\,} , that satisfies: a kind...
encompassing the use of theta-functions. The Siegelmodular varieties, which describe Siegelmodularforms, are recognised as part of the moduli theory...
modularforms awaited the development of complex manifold theory. Siegelmodularform Hilbert modular surface Jan H. Bruinier: Hilbert modularforms and...
In mathematics, a Siegelmodular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed...
Austrian rower Brauer–Siegel theorem Gelfand–Naimark–Segal construction Siegelmodularform Segal space Newell–Whitehead–Segel equation Siegel zero Chagall (disambiguation)...
In mathematics, a Siegel theta series is a Siegelmodularform associated to a positive definite lattice, generalizing the 1-variable theta function of...
a Klingen Eisenstein series is a Siegelmodularform of weight k and degree g depending on another Siegel cusp form f of weight k and degree r<g, given...
In mathematics, the Siegel operator is a linear map from (level 1) Siegelmodularforms of degree d to Siegelmodularforms of degree d − 1, generalizing...
called Hilbert-Blumenthal forms) were proposed not long after that, though a full theory was long in coming. The Siegelmodularforms, for which G is a symplectic...
two Siegelmodularforms to another Siegelmodularform. Miyawaki conjectured the existence of this lift for the case of degree 3 Siegelmodularforms, and...
Siegel Eisenstein series (sometimes just called an Eisenstein series or a Siegel series) is a generalization of Eisenstein series to Siegelmodular forms...
space of abelian varieties, such as the Siegelmodular variety. This is the problem underlying Siegelmodularform theory. See also Shimura variety. Using...
constants are modularforms, or more generally may be an element of a Siegel upper half plane in which case the theta constants are Siegelmodularforms. The theta...
the Weierstrass ℘ function, and Fourier–Jacobi coefficients of Siegelmodularforms of genus 2. Examples with more than two variables include characters...
In mathematics, the Ikeda lift is a lifting of modularforms to Siegelmodularforms. The existence of the lifting was conjectured by W. Duke and Ö. Imamoḡlu...
In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form J of degree 4 and weight 8, introduced by Friedrich Schottky (1888, 1903)...
a generalization of the Siegelmodular group, and has the same relation to polarized abelian varieties that the Siegelmodular group has to principally...
very demanding. And on the side of modularforms, there were examples such as Hilbert modularforms, Siegelmodularforms, and theta-series. There are a number...
formula, was fully expressed by Hiroshi Umemura in 1984, who used Siegelmodularforms in place of the exponential/elliptic transcendents, and replaced...
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