Eisenstein series

Eisenstein series ( after the German mathematician Gotthold Eisenstein ) are different series from the theory of modular forms and automorphic forms.

Holomorphic Eisenstein series

Eisenstein series on the space of grid

Be two complex numbers. The grating generated by and is

The Eisenstein series of weight to the grid in the infinite series of the form

These series are absolutely convergent for, for k odd.

Eisenstein series on the upper half-plane

The investigation of the Eisenstein series can be restricted without loss of generality on the grid form with, because for a lattice with basis is always valid:

And because the base can be chosen such that the following holds, you can see the Eisenstein series of each grid compute as soon as one knows them for those with base. For the latter, we also write for short:

One can thus regarded as a function on the upper half-plane, the Eisenstein series.

The Eisenstein series is a modular form of weight to the group, that is, for having valid

For polynomials with rational coefficients in and, ie, It is the recursion formula:

Especially for n = 4 follows from this and by a comparison of the coefficients of the Fourier developments (see below) the remarkable number theory ( Hurwitz identity, after Adolf Hurwitz ):

Is

The sum of the kth powers of the divisors of n This formula can also elementary ( ie not functional theory) prove.

Fourier expansion

The Eisenstein series can be developed in a Fourier series:

Here is the Riemann zeta function.

Related to elliptic functions

It should be and. Then the Weierstrass ℘ - function for the lattice satisfies the differential equation

Conversely, there are at any elliptic curve over

A grid with and. The elliptic curve is then parameterized by

With. In particular, every elliptic curve is homeomorphic to a torus.