These are the notes for Math 679, University of Michigan, Winter 1996, exactly as they were handed out during the course except for some minor corrections. Please send comments and corrections to me at [email protected] using " Math679 " as the subject.

This book is about elliptic curves and modular functions, two topics that are intimately related in both accidental and essential ways. As emphasized by André Weil in his magisterial historical introduction to contemporary number theory [W], the arithmetic study of elliptic curves is, in spite of the clear reference to the integral calculus in the adjective elliptic, in many respects antecedent...

Z/nZ (n = ∏k i=1 pi ei ) are defined by extending anomalous elliptic curves over a prime filed Fp. They have n points over a ring Z/nZ and pi points over Fpi for all pi. We generalize Satoh-Araki-Smart algorithm [10], [11] and Rück algorithm [9], which solve a discrete logarithm problem over anomalous elliptic curves. We prove that a “discrete logarithm problem over super-anomalous elliptic cur...

We prove that for any given group Zm⊕Zn, where m divides n, and any rational elliptic curve, for a positive density of the rational primes p ∈ P, Zm ⊕ Zn is isomorphic to a subgroup of E(Fp). Our methods are effective and we demonstrate how to construct elliptic curves such that for a large density of the primes p, the given group is isomorphic to a subgroup of E(Fp). We show that for some grou...

Let K be a number field, Galois over Q. A Q-curve over K is an elliptic curve over K which is isogenous to all its Galois conjugates. The current interest in Q-curves, it is fair to say, began with Ribet’s observation [27] that an elliptic curve over Q̄ admitting a dominant morphism from X1(N) must be a Q-curve. It is then natural to conjecture that, in fact, all Q-curves are covered by modular ...

In this paper we introduce two new generic sidechannel attacks on scalar blinding of elliptic curves where the order of the base point is close to a power of 2. These attacks are in particu­ lar relevant for elliptic curves over special prime fields where the prime is ’almost’ a power of 2. As in the papers [9, 10] we assume that some side-channel attack has allowed the determina­ tion of the b...

The rank conjecture says that rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. We prove the conjecture for elliptic curves E(K) over a number field K. As a corollary, one gets a simple estimate for the rank in terms of the length of period of a continued fraction attached to the E(K). As an illustration, we consider a family of elliptic...