Supersingular elliptic curves, theta series and weight two modular forms

Elliptic Curves and Modular Forms

This is an exposition of some of the main features of the theory of elliptic curves and modular forms.

Elliptic Curves , Arithmetic Geometry , and Modular Forms

Constructing Supersingular Elliptic Curves

We give an algorithm that constructs, on input of a prime power q and an integer t, a supersingular elliptic curve over Fq with trace of Frobenius t in case such a curve exists. If GRH holds true, the expected run time of our algorithm is e O((log q)). We illustrate the algorithm by showing how to construct supersingular curves of prime order. Such curves can readily be used for pairing based c...

ELLIPTIC CURVES AND MODULAR FORMS Contents

1. January 21, 2010 2 1.1. Why define a curve to be f rather than V (f) ⊂ P(k)? 3 1.2. Cubic plane curves 3 2. January 26, 2010 4 2.1. A little bit about smoothness 4 2.2. Weierstrass form 5 3. January 28, 2010 6 3.1. An algebro-geometric description of the group law in terms of divisors 6 3.2. Why are the two group laws the same? 7 4. February 2, 2010 7 4.1. Overview 7 4.2. Uniqueness of Weier...

Elliptic Curves , Modular Forms and Related

Nick Andersen (University of California, Los Angeles) Kloosterman sums and Maass cusp forms of half-integral weight ABSTRACT: Kloosterman sums play an important role in modern analytic number theory. I will give a brief survey of what is known about the classical Kloosterman sums and their connection to Maass cusp forms of weight 0. I will then talk about recent progress toward bounding sums of...