scholarly journals Theta Functions and Abelian Varieties over Valuation Fields of Rank one I

1962 ◽  
Vol 20 ◽  
pp. 1-27 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Imaginary Part ◽  
Classical Theory ◽  
Theta Function ◽  
Abelian Varieties ◽  
Theta Functions ◽  
Positive Definite ◽  
Complex Matrix ◽  
Rank One ◽  
Symmetric Complex

We shall denote by the Z-module of integral vectors of dimension r, by T a symmetric complex matrix with positive definite imaginary part and by g the variable vector. If we put and the fundamental theta function is expressed in the form: as a series in q and u. Other theta functions in the classical theory are derived from the fundamental theta function by translating the origin and making sums and products, so these theta functions are also expressed in the form: as series of q and u. Moreover the coefficients in the relations of theta functions are also expressed in the form: as series in q.

scholarly journals A decomposition theorem on differential polynomials of theta functions

1984 ◽  
Vol 96 ◽  
pp. 113-126 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Entire Function ◽  
Imaginary Part ◽  
Theta Function ◽  
Theta Functions ◽  
Decomposition Theorem ◽  
Positive Definite ◽  
Complex Variables ◽  
Differential Polynomials ◽  
Symmetric Complex ◽  
The Difference

Let τ = (τij) be a symmetric complex g = g matrix with the positive definite imaginary part. A theta function of level n means an entire function f(z) in g complex variables z = (zl …, zg) satisfying the difference relations:

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.

scholarly journals Higher depth mock theta functions and q-hypergeometric series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Males ◽  
Andreas Mono ◽  
Larry Rolen
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

scholarly journals Topology of Subvarieties of Complex Semi-abelian Varieties

2020 ◽  
Author(s):  
Yongqiang Liu ◽  
Laurentiu Maxim ◽  
Botong Wang
Keyword(s):  
Euler Characteristic ◽  
Morse Theory ◽  
Characteristic Property ◽  
Abelian Varieties ◽  
Betti Numbers ◽  
Local Systems ◽  
Affine Manifolds ◽  
Cyclic Covers ◽  
Rank One ◽  
Generic Vanishing

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

scholarly journals Minimal theta functions

1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Functional Equation ◽  
Quadratic Form ◽  
Zeta Function ◽  
Theta Functions ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Epstein Zeta Function ◽  
Image Position

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

scholarly journals An alternate circular summation formula of theta functions and its applications

2012 ◽  
Vol 6 (1) ◽  
pp. 114-125 ◽  
Author(s):  
Jun-Ming Zhu
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Summation Formula ◽  
Theta Function Identities

We prove a general alternate circular summation formula of theta functions, which implies a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two formulaes for (q; q)2n?.

Absolute Diagonal Scaling Preconditioner of COCG Method for Symmetric Complex Matrix

2007 ◽  
pp. 113-117
Author(s):  
Moethuthu ◽  
Noriko Okamoto ◽  
Akira Shiode ◽  
Seiji Fujino ◽  
Toru Otsuru
Keyword(s):  
Complex Matrix ◽  
Diagonal Scaling ◽  
Symmetric Complex

On Ramanujan's general theta function and a generalization of the Borweins' cubic theta functions

2015 ◽  
Vol 08 (01) ◽  
pp. 1550002
Author(s):  
Chandrashekar Adiga ◽  
Nasser Abdo Saeed Bulkhali
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Special Case

The Borwein brothers have introduced and studied three cubic theta functions. Many generalizations of these functions have been studied as well. In this paper, we introduce a new generalization of these functions and establish general formulas that are connecting our functions and Ramanujan's general theta function. Many identities found in the literature follow as a special case of our identities. We further derive general formulas for certain products of theta functions.

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