Counting solutions of a quadratic equation - LECTURE 2

  • ABSTRACT: Let f be a given nonsingular quadratic form in n variables with integer coefficients. Let m be a non-negative integer, either 0 or tending to infinity. We are interested in giving an asymptotic formula for the number of integer solutions of
    f(x) = m
    in a box PB, where the box B is fixed and the dilation factor P tends to infinity. We make the convention that P is the square root of m, if m is not 0. The first lecture describes the Hardy -Littlewood method and its application to this problem when n is at least 5. Kloosterman's refinement of the method is then descibed, with some background about Kloosterman sums. This takes care of n = 4 and m positive.

    In the second lecture the discussion shifts to Heath-Brown's version of the circle method. The problem can now be tackled when m = 0 and n is either 3 or 4. In the latter case, it makes a remarkable difference whether or not the determinant of f is a square.

    In conclusion, there is a short discussion of the approach to ternary positive definite f that began with work of Iwaniec in 1986.

  • LECTURE NOTES

The url of this page is http://qseries.org/fgarvan/quadformsconf/workshop-program/baker2.html.
Created by F.G. Garvan (fgarvan@math.ufl.edu) on Saturday, February 28, 2009.
Last update made Sat Mar 7 20:21:47 EST 2009.

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