Modular Forms - A Master's Thesis
Author
Larsen, Mikkel Højlund
Term
4. term
Education
Publication year
2020
Submitted on
2020-06-02
Pages
79
Abstract
Dette speciale undersøger modulære former for at udlede en eksplicit formel for antallet af repræsentationer af et ikke-negativt heltal n som en sum af fire heltalskvadrater. Udgangspunktet er Jacobi theta-funktionen, hvor koefficienterne i θ(z)^4 svarer til r(n,4). Specialet introducerer modulære former for SL2(Z), Eisenstein-rækker og deres Fourier-udvidelser, og udvider til kongruensdelgrupper, især Γ0(4). Det vises, at θ(z)^4 er en modulær form af vægt 2 for Γ0(4), at rummet M2(Γ0(4)) er todimensionalt med basiselementerne G2(z)−2G2(2z) og G2(z)−4G2(4z), og at θ(z)^4 kan skrives som −(1/π^2)·(G2(z)−4G2(4z)). Ved sammenligning af Fourier-koefficienter opnås en enkel divisorformel: r(n,4) er 8 gange summen af alle positive divisorer d af n, der ikke er delelige med 4. Afslutningsvis skitseres, hvordan tilsvarende metoder kan anvendes til lige potenser af θ(z) for at studere repræsentationer som sum af k kvadrater.
This thesis uses modular forms to derive an explicit formula for the number of representations of a nonnegative integer n as a sum of four integer squares. The starting point is the Jacobi theta function, whose coefficients in θ(z)^4 equal r(n,4). The work introduces modular forms for SL2(Z), Eisenstein series and their Fourier expansions, and then considers congruence subgroups, in particular Γ0(4). It is shown that θ(z)^4 is a weight-2 modular form for Γ0(4), that the space M2(Γ0(4)) is two-dimensional with basis G2(z)−2G2(2z) and G2(z)−4G2(4z), and that θ(z)^4 can be written as −(1/π^2)·(G2(z)−4G2(4z)). Comparing Fourier coefficients yields a concise divisor formula: r(n,4) equals 8 times the sum of all positive divisors d of n that are not divisible by 4. Finally, the thesis outlines how similar methods apply to even powers of θ(z) to study representations as sums of k squares.
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