Lecture notes for a series of talks given in Bonn, June 2015. Most of the topics covered touched in one way or another on the role of power operations in elliptic cohomology. In June of 2015 I gave a series of six lectures (the Felix Klein lectures) in Bonn. These are some of my lecture notes for those talks. I had hoped to polish them more carefully, but that hasn’t happend yet, and at this po...

In their seminal paper, Miyaji, Nakabayashi and Takano [12] describe a simple method for the creation of elliptic curves of prime order with embedding degree 3, 4, or 6. Such curves are important for the realisation of pairing-based cryptosystems on ordinary (non-supersingular) elliptic curves. We provide an alternative derivation of their results, and extend them to allow for the generation of...

An ongoing challenge in cryptography is to find groups in which the DLP is computationally infeasible, that is, for which the best known attack is exponential in log(N). Such a group can be used as the setting for many cryptographic protocols, from Diffie-Hellman key exchange to El Gamal encryption ([14], 159). The most prominent example, first proposed in 1985, is a subgroup of points of an el...

results in more than 30.000 hits, ranging from articles in mathematics and computer science to books and implementations and applications of the algorithms. There are a lot of different algorithms and also different ways to attack them, many of them involving a huge amount of theory. Moreover, the fact that still there is not an accepted standard on elliptic curve cryptography benefits the exis...

A detailed study is made of super elliptic curves, namely super Riemann surfaces of genus one considered as algebraic varieties, particularly their relation with their Picard groups. This is the simplest setting in which to study the geometric consequences of the fact that certain cohomology groups of super Riemann surfaces are not freely generated modules. The divisor theory of Rosly, Schwarz,...

The purpose of these notes is to provide a quick introduction to the moduli of elliptic curves. There are many excellent and thorough references on the subject, ranging from the slightly archaic [Igu59] and [Shi94] to the more difficult [KM85] and [DR73]. Brian’s forthcoming book on the Ramanujan conjecture also covers some of this material and includes a careful comparison of the transcendenta...

Elliptic curves also figured prominently in the recent proof of Fermat's Last Theorem by Andrew Wiles. Originally pursued for purely aesthetic reasons, elliptic curves have recently been utilized in devising algorithms for factoring integers, primality proving, and in public-key cryptography. In this article, we aim to give the reader an introduction to elliptic curve cryptosystems, and to demo...

This paper gives a general survey of ranks of elliptic curves over the field of rational numbers. The rank is a measure of the size of the set of rational points. The paper includes discussions of the Birch and SwinnertonDyer Conjecture, the Parity Conjecture, ranks in families of quadratic twists, and ways to search for elliptic curves of large rank.

The discriminant Define b2 := a1 + 4a2, b4 := 2a4 + a1a3, b6 := a3 + 4a6, b8 := a1a6 + 4a2a6 − a1a3a4 + a2a3 − a4. Note that if char(K) 6= 2, then we can perform the coordinate transformation y 7→ (y− a1x− a3)/2 to arrive at an equation y = 4x + b2x +2b4x+ b6. Now for any Weierstrass equation we define its discriminant ∆ := −b2b8 − 8b4 − 27b6 + 9b2b4b6. Proposition 1. A curve C/K given by a Wei...

In recent years, the question of whether the ranks of elliptic curves defined over Q are unbounded has garnered much attention. One can create refined versions of this question by restricting one’s attention to elliptic curves over Q with a certain algebraic structure, e.g., with a rational point of a given order. In an attempt to gather data about such questions, we look for examples of ellipt...