531 research outputs found
Approximate computations with modular curves
This article gives an introduction for mathematicians interested in numerical
computations in algebraic geometry and number theory to some recent progress in
algorithmic number theory, emphasising the key role of approximate computations
with modular curves and their Jacobians. These approximations are done in
polynomial time in the dimension and the required number of significant digits.
We explain the main ideas of how the approximations are done, illustrating them
with examples, and we sketch some applications in number theory
Non-canonical extension of theta-functions and modular integrability of theta-constants
This is an extended (factor 2.5) version of arXiv:math/0601371 and
arXiv:0808.3486. We present new results in the theory of the classical
-functions of Jacobi: series expansions and defining ordinary
differential equations (\odes). The proposed dynamical systems turn out to be
Hamiltonian and define fundamental differential properties of theta-functions;
they also yield an exponential quadratic extension of the canonical
-series. An integrability condition of these \odes\ explains appearance
of the modular -constants and differential properties thereof.
General solutions to all the \odes\ are given. For completeness, we also solve
the Weierstrassian elliptic modular inversion problem and consider its
consequences. As a nontrivial application, we apply proposed techni\-que to the
Hitchin case of the sixth Painlev\'e equation.Comment: Final version; 47 pages, 1 figure, LaTe
Sigma, tau and Abelian functions of algebraic curves
We compare and contrast three different methods for the construction of the
differential relations satisfied by the fundamental Abelian functions
associated with an algebraic curve. We realize these Abelian functions as
logarithmic derivatives of the associated sigma function. In two of the
methods, the use of the tau function, expressed in terms of the sigma function,
is central to the construction of differential relations between the Abelian
functions.Comment: 25 page
Higher dimensional 3-adic CM construction
We find equations for the higher dimensional analogue of the modular curve
X_0(3) using Mumford's algebraic formalism of algebraic theta functions. As a
consequence, we derive a method for the construction of genus 2 hyperelliptic
curves over small degree number fields whose Jacobian has complex
multiplication and good ordinary reduction at the prime 3. We prove the
existence of a quasi-quadratic time algorithm for computing a canonical lift in
characteristic 3 based on these equations, with a detailed description of our
method in genus 1 and 2.Comment: 23 pages; major revie
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