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Jacobi theta function
Redirected from "Jacobi theta-functions".
Contents
Context
Theta functions
Complex geometry
Arithmetic geometry
number theory
number
- natural number, integer number, rational number, real number, irrational number, complex number, quaternion, octonion, adic number, cardinal number, ordinal number, surreal number
arithmetic
arithmetic geometry, function field analogy
Arakelov geometry
Contents
Idea
The fundamental example of theta functions is the Jacobi theta function given by
As a variable of two arguments, this is actually a Jacobi form. These are the local coordinate expressions of the the covariantly constant sections of the Hitchin connection (for circle gauge group) on the moduli space of elliptic curves (Hitchin 90, remark 4.12). See there for more and see at theta function for the general idea.
At the function
is what in number theory is often just called “the theta function”. This is the one whose Mellin transform is the Riemann zeta function, see at Riemann zeta function – Relation to Jacobi theta function
Properties
Functional equation and Reciprocity
By the Poisson summation formula the number-theoretic theta function satisfies the following functional equation:
Under the Mellin transform this implies the functional equation of the Riemann zeta function, see at Riemann zeta function – Functional equation.
It also provides an analytic proof of the Landsberg-Schaar relation?
where and are arbitrary positive integers. To prove it, apply theta reciprocity to , , and then let .
This reduces to the formula for the quadratic Gauss sum when :
(where is an odd prime). From this, it’s not hard to deduce Gauss’s “golden theorem”.
quadratic reciprocity: for odd primes and .
See e.g. (Karlsson).
some of this material from this MO discussion
context/function field analogy | theta function | zeta function (= Mellin transform of ) | L-function (= Mellin transform of ) | eta function | special values of L-functions |
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physics/2d CFT | partition function as function of complex structure of worldsheet (hence polarization of phase space) and background gauge field/source | analytically continued trace of Feynman propagator | analytically continued trace of Feynman propagator in background gauge field : | analytically continued trace of Dirac propagator in background gauge field | regularized 1-loop vacuum amplitude / regularized fermionic 1-loop vacuum amplitude / vacuum energy |
Riemannian geometry (analysis) | | zeta function of an elliptic differential operator | zeta function of an elliptic differential operator | eta function of a self-adjoint operator | functional determinant, analytic torsion |
complex analytic geometry | section of line bundle over Jacobian variety in terms of covering coordinates on | zeta function of a Riemann surface | Selberg zeta function | | Dedekind eta function |
arithmetic geometry for a function field | | Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry) | | | |
arithmetic geometry for a number field | Hecke theta function, automorphic form | Dedekind zeta function (being the Artin L-function for the trivial Galois representation) | Artin L-function of a Galois representation , expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps) | | class number regulator |
arithmetic geometry for | Jacobi theta function ()/ Dirichlet theta function ( a Dirichlet character) | Riemann zeta function (being the Dirichlet L-function for Dirichlet character ) | Artin L-function of a Galois representation , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function | | |
References
Due to Carl Jacobi.
Review includes
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Wikipedia, Jacobi theta-function
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section 9 in Analytic theory of modular forms (pdf)
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Anders Karlsson, Applications of heat kernels on abelian groups: , quadratic reciprocity, Bessel integrals (psd)
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Nigel Hitchin, Flat connections and geometric quantization, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (Euclid)
Last revised on April 26, 2020 at 10:41:41.
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