Eisenstein Series In Modular Form (Essay Sample)

Eisenstein Series In Modular Form
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Introduction to modular form
A modular form is an analytic satisfying a particular type of functional equation on the upper half- plane with respect to the group action of a modular group and also satisfies a given growth condition.
Definition of terms
Modular group; this is a group of all matrices with a, b, c, d   and ad-bc= 1
For instance;
S= and T= this are both elements of . For a given positive r integer, the modular group is given by;
Congruence subgroup; is by example a 2×2 simple matrix that can be inverted with a determinant of 1 and off-diagonal entries that are even.
Cusp form; this is a given kind of modular form having a constant zero coefficient in the Fourier series expansion. A cusp form can be distinguished in the modular forms due to its disappearing Fourier series expansion.
Mass forms; these are Laplacian real analytic Eigen functions that do not necessarily have to be holomorphic.
Hilbert modular forms; these are functions in the n variable with each complex number in the upper half plane and satisfies a modular relation for a 2 × 2 matrices with the entries being in a totally real number field.
Siegel modular forms; these are related to abelian varieties in the same way that elliptic modular forms are related to elliptic curves.
Jacobi forms; these are a mixture of elliptic functions and modular forms. They tend to have an arithmetic theory that is similar to the usual modular form theory.
Automorphic forms; these serves the purpose of extending the notion of modular forms to general Lie groups.
Eisenstein series
The Eisenstein series was named after a German, Gothhold Eisenstein who was a mathematician. These are specific modular forms with expansions of the infinite series which can be directly written down. Eisenstein series was originally defined for the modular group, and could be generalized in the automorphic forms theory.
Define a holomorphic Eisenstein series, with a complex positive number T, G2k(Ï„) of weight 2k, where k is greater than or equal to 2 and is an integer, by using the following series.
This series converges into a holomorphic function of T, at the upper half plane. Its Fourier
Eisenstein series is ideally a modular form having the key property as its SL (2, Z) invariance. Clearly if a, b, c, dz and ad-bc=1 then
G2k thus becomes a modular form of weight 2k assuming that k ≥ 2 otherwise it would be illegitimate to change the order of summation, while SL (2, Z) invariance would not hold.
The first two series of the Eisenstein series gives the modular invariants g2 and g3 of an elliptic curve as:
For the modular group, any holomorphic modular form can be written as a polynomial in G4 and G6 and the high order of G2k's can be written in terms of G4 and G6 via a recurrence relation
If we let dk = (2k+3) k!G2k+4, Then satisfies the relation;
For all the n ≥ 0. becomes the binomial coefficient whereas:
and and .
Occurs within the series expansion of the weierstrass’s elliptic functions.
Fourier series
When defining the Fourier series of the Einstein series becomes
Where the coefficients of C2k are given by
Here Bn Are the Bernoulli’s numbers .
ζ (z) is the Riemann’s zeta function and σp (n) becomes the divisor sum of the equation. This is the summation of the powers of the divisors of n.
The lambert series will be the summation over q, thus we will have
Working with the q expansion of the Eisenstein series for arbitrary complex |q|≤1 the notation below is normally introduced.
Eisenstein series identities
Given that , let
We define;
And are the alternative notations representing Jacobi theta functions. Thus;
Hence;
The expression related to the modular discriminant will be given by;
Imply that;
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The products of the Eisenstein series
Eisenstein series makes the most precise examples of modulus forms, especially for the modular group SL (2, Z).Different products of the Eisenstein series with the modular fo...