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 On Abelian Varieties

1953 ◽  
Vol 6 ◽  
pp. 151-170 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
General Theory ◽  
Abelian Varieties ◽  
Theta Functions ◽  
Analytic Structure ◽  
Fundamental Theorems

In a Bourbaki seminary note, La Théorie des Fonctions Thêta, A. Weil has discussed two fundamental theorems of the general theory of Theta functions. The first, due to H. Poincaré, was proved very skilfully in the note by means of harmonic integrals on a torus and the second, due to Frobenius, was treated by the systematic use of the notion of analytic structure.

scholarly journals On the Defining Equations of Abelian Varieties

1967 ◽  
Vol 30 ◽  
pp. 143-162 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Present Article ◽  
Abelian Varieties ◽  
Theta Functions

This article is a continuation and completion of our study in the previous one [1] in which we gave the defining equations of abelian varieties projectively embedded by theta functions of level three. These defining equations are nothing else the canonical generators of theta relations for theta functions with ⅓—characteristics. In the present article the canonical generators of 3 theta relations will be given for the wider class of theta functions with rational characteristics.

scholarly journals Definability of restricted theta functions and families of abelian varieties

2013 ◽  
Vol 162 (4) ◽  
pp. 731-765 ◽  
Author(s):  
Ya’acov Peterzil ◽  
Sergei Starchenko
Keyword(s):  
Abelian Varieties ◽  
Theta Functions

scholarly journals Computing isogenies between abelian varieties

2012 ◽  
Vol 148 (5) ◽  
pp. 1483-1515 ◽  
Author(s):  
David Lubicz ◽  
Damien Robert
Keyword(s):  
Abelian Variety ◽  
Time Complexity ◽  
Abelian Varieties ◽  
Theta Functions ◽  
Null Point ◽  
Constant Number ◽  
Weil Pairing ◽  
Base Field ◽  
Rational Expression ◽  
Higher Dimensional

AbstractWe describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let A be an abelian variety of dimension g defined over a field of odd characteristic. Our algorithm comprises two principal steps. First, given a theta null point for A and a subgroup K isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian variety A/K. Then, from the knowledge of a theta null point of A/K, we present an algorithm to obtain a rational expression for an isogeny from A to A/K. The algorithm that results from combining these two steps can be viewed as a higher-dimensional analog of the well-known algorithm of Vélu for computing isogenies between elliptic curves. In the case where K is isomorphic to (ℤ/ℓℤ)g for ℓ∈ℕ*, the overall time complexity of this algorithm is equivalent to O(log ℓ) additions in A and a constant number of ℓth root extractions in the base field of A. In order to improve the efficiency of our algorithms, we introduce a compressed representation that allows us to encode a point of level 4ℓ of a g-dimensional abelian variety using only g(g+1)/2⋅4g coordinates. We also give formulas for computing the Weil and commutator pairings given input points in theta coordinates.

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