Convolution and Square in Abelian Groups I

Abstract

We prove that the functional equation $f\mathrm{}\star \mathrm{}f(2 t)=\lambda f{(t)}^2$, for t in $\mathbb{Z}/d\mathbb{Z}$ with d odd, admits a nonzero solution f if $\lambda =\sqrt{a}\mathrm{}+\mathrm{}i\sqrt{b}$ with a, b positive integers such that $a\mathrm{}+\mathrm{}b\mathrm{}=\mathrm{}d$ and $a\mathrm{}\equiv \frac{{(d+1)}^2}{4}$ mod 4. The proof involves theta functions on elliptic curves with complex multiplication.

Keywords:

• functional equation
• convolution
• cyclic groups
• elliptic curves
• complex multiplication
• theta functions
• modular curve

2010 Mathematics Subject Classification:

• Primary: 11F03
• Secondary: 11F27

Acknowledgments

I thank Gérard Laumon, Samuel Lelièvre and Emmanuel Ullmo for enlighting discussions on this project.

Notes

1 With other “classical” notations for theta functions as in [3], one has the equalities, ${\theta }_{[0]}(z,\tau )={\theta }_{0,0}(2z,2\tau )=\theta \left[\text{}\begin{array}{c}0\\ 0\end{array}\text{}\right](2z,2\tau )$ and ${\theta }_{[1]}(z,\tau )={\theta }_{1,0}(2z,2\tau )=\theta \left[\text{}\begin{array}{c}1/2\\ 0\end{array}\text{}\right](2z,2\tau ).$