Period matrix of the Jacobian variety of a curve

In summary, an algebraic variety, X, is a smooth algebraic manifold specified as the zero set of a known polynomial. The period matrix of X, or more precisely, the Jacobian variety of the curve, can be computed numerically if the explicit construction is followed.
  • #1
GogoJS
3
0
Consider an algebraic variety, [itex]X[/itex] which is a smooth algebraic manifold specified as the zero set of a known polynomial.

I would appreciate resource recommendations preferably or an outline of approaches as to how one can compute the period matrix of [itex]X[/itex], or more precisely, of the Jacobian variety of the curve.

My motivation is I am studying the Kawazumi-Zhang invariant (related to the Faltings invariant) of Riemann surfaces, which can be expressed as a Fourier series in the period matrix, and thus if I can evaluate the period matrix [itex]\Omega[/itex] of the algebraic manifold, then I can compute its Kawazumi-Zhang invariant, numerically.
 
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  • #2
Not sure whether it helps, but I've found

https://www.amazon.com/dp/3540102906/?tag=pfamazon01-20

which deals with algorithms on algebraic varieties in the sense, that it is less about the mathematics but more about actual algorithms. In the section which is about the Jacobian (p. 106 ff.) Davenport refers to

https://www.amazon.com/dp/1614276129/?tag=pfamazon01-20

However, the latter might be more about boundaries than actual algorithms. A suspicion I have due to the year of its original publication (1959). E.g. the first one proves an upper bound on ##|Jac(C)|_K##. In any case Davenport provides a long list of references, which might be the starting point for further searches.
 
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  • #3
The question of explicit computation of "abelian integrals", as they are usually called, is one I know little about, but the following paper of Gross may be relevant. It contains an appendix by Rohrlich treating such "computations" in a special case, and also references a paper of Weil on this topic, [W3] in the bibliography of Gross's paper.

http://s3.amazonaws.com/academia.edu.documents/46450483/bf0139027320160613-7495-1pri9m8.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1485031611&Signature=ZmYs9hrDdN7BRFB8jNp71o73YOI%3D&response-content-disposition=inline%3B%20filename%3DOn_the_periods_of_abelian_integrals_and.pdf

Mumford's Lectures on Theta functions, volume II, gives an explicit construction, also following Weil, of the Jacobi variety of a hyperelliptic curve, (the projective closure of the locus) defined by an equation of form Y^2 = f(X), for a polynomial f. Maybe it could be useful.

https://www.amazon.com/dp/0817645691/?tag=pfamazon01-20
 
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FAQ: Period matrix of the Jacobian variety of a curve

What is the period matrix of the Jacobian variety of a curve?

The period matrix of the Jacobian variety of a curve is a matrix that encodes important geometric and arithmetic information about the curve. It is a fundamental tool in the study of algebraic curves and their properties.

How is the period matrix computed?

The period matrix can be computed using the Riemann theta function. This is a special function that is associated with the curve and is used to transform the curve's geometry into a matrix form.

What is the significance of the period matrix?

The period matrix is significant because it contains information about the topology, geometry, and arithmetic of the curve. It is a powerful tool in the study of algebraic curves and their applications in areas such as cryptography and coding theory.

Can the period matrix reveal if a curve is singular?

Yes, the period matrix can reveal if a curve is singular. In particular, the number of zeros of the Riemann theta function can indicate the number of singular points on the curve.

How is the period matrix used in cryptography?

The period matrix is used in cryptography for its ability to encode information about the curve, making it difficult for an attacker to extract sensitive data. It is also used in the construction of secure elliptic curve cryptosystems.

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