Book Review: Algebraic curves over finite fields

Workshop on Algebraic Curves Over Finite Fields

Let L(t) = 1+a1t+ · · ·+a2gt be the numerator of the zeta function of an algebraic curve C defined over the finite field Fq of genus g. We show that the coefficients ar of L(t) satisfy certain inequalities. Conversely, for any integers a1, . . . , am satisfying these inequalities and all sufficiently large integers g there exist curves of genus g, whose L-polynomial satisfies the following cong...

Algebraic complexities and algebraic curves over finite fields

We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear c...

Elliptic Curves over Finite Fields

In this chapter, we study elliptic curves defined over finite fields. Our discussion will include the Weil conjectures for elliptic curves, criteria for supersingularity and a description of the possible groups arising as E(Fq). We shall use basic algebraic geometry of elliptic curves. Specifically, we shall need the notion and properties of isogenies of elliptic curves and of the Weil pairing....

Elliptic Curves over Finite Fields

Counting curves over finite fields

Article history: Received 25 August 2014 Received in revised form 10 September 2014 Accepted 18 September 2014 Available online 4 November 2014 Communicated by H. Stichtenoth MSC: 11G20 10D20 14G15 14H10