scholarly journals Algebraic Geometry and Theta Functions

Nature ◽  
1930 ◽  
Vol 125 (3160) ◽  
pp. 775-775
Author(s):  
H. T. H. P.
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions

scholarly journals Theta Surfaces

2020 ◽  
Author(s):  
Daniele Agostini ◽  
Türkü Özlüm Çelik ◽  
Julia Struwe ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions ◽  
Minkowski Sum ◽  
Zero Set ◽  
Analytic Surface ◽  
Numerical Algebraic Geometry ◽  
Abelian Integrals ◽  
Space Curves ◽  
Quartic Curves ◽  
Special Plane

Abstract A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

scholarly journals Algebraic geometry and theta functions

1935 ◽  
Vol 42 (1) ◽  
pp. A9-A9
Author(s):  
H. Hornich
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions

30 years of finite-gap integration theory

2007 ◽  
Vol 366 (1867) ◽  
pp. 837-875 ◽  
Author(s):  
Vladimir B Matveev
Keyword(s):  
Algebraic Geometry ◽  
Riemann Surfaces ◽  
Theta Functions ◽  
Theoretical Physics ◽  
Integration Theory ◽  
Difference Operators ◽  
Strong Impact ◽  
Riemann Theta Functions ◽  
Compact Riemann Surfaces ◽  
Modern Mathematics

The method of finite-gap integration was created to solve the periodic KdV initial problem. Its development during last 30 years, combining the spectral theory of differential and difference operators with periodic coefficients, the algebraic geometry of compact Riemann surfaces and their Jacobians, the Riemann theta functions and inverse problems, had a strong impact on the evolution of modern mathematics and theoretical physics. This article explains some of the principal historical points in the creation of this method during the period 1973–1976, and briefly comments on its evolution during the last 30 years.

scholarly journals On the Coble quartic and Fourier–Jacobi expansion of theta relations

2015 ◽  
Vol 26 (02) ◽  
pp. 1550019
Author(s):  
Francesco Dalla Piazza ◽  
Riccardo Salvati Manni
Keyword(s):  
Algebraic Geometry ◽  
American Mathematical Society ◽  
Theta Functions ◽  
Mathematical Society ◽  
Jacobi Expansion ◽  
The Ideal

In [Q. Ren, S. Sam, G. Schrader and B. Sturmfels, The universal Kummer threefold, Experiment Math.22(3) (2013) 327–362], the authors conjectured equations for the universal Kummer variety in genus 3 case. Although, most of these equations are obtained from the Fourier–Jacobi expansion of relations among theta constants in genus 4, the more prominent one, Coble's quartic, cf. [A. Coble, Algebraic Geometry and Theta Functions, American Mathematical Society Colloquium Publications, Vol. 10 (American Mathematical Society, 1929)] was obtained differently, cf. [S. Grushevsky and R. Salvati Manni, On Coble's quartic, preprint (2012), arXiv:1212.1895] too. The aim of this paper is to show that Coble's quartic can be obtained as Fourier–Jacobi expansion of a relation among theta-constants in genus 4. We get also one more relation that could be in the ideal described in [Experiment Math.22(3) (2013) 327–362].

scholarly journals On Theta Functions and Abelian Varieties over Valuation Fields of Rank One: (II) Theta Functions and Abelian Functions of Characteristic p(>0)

1962 ◽  
Vol 21 ◽  
pp. 231-250 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Algebraic Geometry ◽  
Abelian Varieties ◽  
Theta Functions ◽  
Elliptic Functions ◽  
Characteristic P ◽  
Canonical Systems ◽  
Rank One ◽  
Abelian Functions ◽  
Explicit Expressions

It may safely said that one of the most important problems in modern algebraic geometry is to elevate theory of abelian functions to the same level as theory of elliptic functions beyond the modern formulation of classical results. Being concerned in such a problem, we feel that one of the serious points is the lack of knowladge on the explicit expressions of abelian varieties and their law of compositions by means of their canonical systems of coordinates: Such expressions correspond to the cubic relation of Weierstrass’ -functions and their addition formulae in theory of elliptic functions.

Algebraic Geometry and Theta Functions

1929 ◽  
Vol 14 (204) ◽  
pp. 582
Author(s):  
H. P. H. ◽  
A. B. Coble
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions
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