Math7280 : Modular forms and elliptic curves

Modular forms are certain functions on the upper half complex plane. Usually they are given as q expansions, where $ q=\exp(2\pi iz)$ and z is in the upper half plane. The coefficients of modular forms often satisfy interesting number theoretic properties. Some examples are:

$\displaystyle \Delta(z) = q \prod_{n=1}^{\infty} \left( 1-q^n\right) ^{24}= \sum_{n=1}^{\infty} \tau(n) q^n$

Amazing fact: Ramanujan showed that for all primes p,

$\displaystyle \tau(p) \equiv 1+p^{11} ~\mod ~ 691.$

Now let an be coefficients of the following modular form:

$\displaystyle \prod_{n=1}^\infty (1-q^{11n})^2(1-q^n)^2=\sum_{n\ge 1}a_n q^n.$

Amazing fact: For all primes p (apart from 11),

$\displaystyle a_p = p -\char93 \{(x,y)\in\mathbb{F}_p \vert y^2 + y = x^3 - x^2 -10x -20\}.$

Here, Fp is a finite field with p elements, and y2 + y = x3 - x2 -10x -20 is an example of an elliptic curve.

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For Spring 2004, by H. Verrill, 26 Oct 03