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A Period matrix of the Jacobian variety of a curve

  1. Jan 21, 2017 #1
    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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  3. Jan 21, 2017 #2

    fresh_42

    Staff: Mentor

    Not sure whether it helps, but I've found

    https://www.amazon.com/Integration-...rds=on+the+integration+of+algebraic+functions

    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/Abelian-Vari...&sr=8-1&keywords=serge+lang+abelian+varieties

    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.
     
    Last edited by a moderator: May 8, 2017
  4. Jan 21, 2017 #3

    mathwonk

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    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=&response-content-disposition=inline; filename=On_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/Tata-Lecture...8759&sr=1-1&keywords=mumford,+theta+functions
     
    Last edited by a moderator: May 8, 2017
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