On isogeny classes of Edwards curves over finite fields

We count the number of isogeny classes of Edwards curves over finite fields, answering a question recently posed by Rezaeian and Shparlinski. We also show that each isogeny class contains a {\em complete} Edwards curve, and that an Edwards curve is isogenous to an {\em original} Edwards curve over \F_q if and only if its group order is divisible by 8 if $q \equiv -1 \pmod{4}$, and 16 if $q \equiv 1 \pmod{4}$. Furthermore, we give formulae for the proportion of d \in \F_q \setminus \{0,1\} for which the Edwards curve $E_d$ is complete or original, relative to the total number of $d$ in each isogeny class.Comment: 27 page

On the Distribution of the Number of Points on Algebraic Curves in Extensions of Finite Fields

Let \cC be a smooth absolutely irreducible curve of genus $g \ge 1$ defined over \F_q, the finite field of $q$ elements. Let # \cC(\F_{q^n}) be the number of \F_{q^n}-rational points on \cC. Under a certain multiplicative independence condition on the roots of the zeta-function of \cC, we derive an asymptotic formula for the number of $n =1, ..., N$ such that (# \cC(\F_{q^n}) - q^n -1)/2gq^{n/2} belongs to a given interval \cI \subseteq [-1,1]. This can be considered as an analogue of the Sato-Tate distribution which covers the case when the curve \E is defined over \Q and considered modulo consecutive primes $p$, although in our scenario the distribution function is different. The above multiplicative independence condition has, recently, been considered by E. Kowalski in statistical settings. It is trivially satisfied for ordinary elliptic curves and we also establish it for a natural family of curves of genus $g=2$.Comment: 14 page

Generalization of a theorem of Carlitz

AbstractWe generalize Carlitzʼ result on the number of self-reciprocal monic irreducible polynomials over finite fields by showing that similar explicit formula holds for the number of irreducible polynomials obtained by a fixed quadratic transformation. Our main tools are a combinatorial argument and Hurwitz genus formula

Decomposing Jacobians of Curves over Finite Fields in the Absence of Algebraic Structure

We consider the issue of when the L-polynomial of one curve over \F_q divides the L-polynomial of another curve. We prove a theorem which shows that divisibility follows from a hypothesis that two curves have the same number of points over infinitely many extensions of a certain type, and one other assumption. We also present an application to a family of curves arising from a conjecture about exponential sums. We make our own conjecture about L-polynomials, and prove that this is equivalent to the exponential sums conjecture.Comment: 20 page

Multiplicative Order of Gauss Periods

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.Comment: 9 page

On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields

AbstractUsing the Stickelberger–Swan theorem, the parity of the number of irreducible factors of a self-reciprocal even-degree polynomial over a finite field will be hereby characterized. It will be shown that in the case of binary fields such a characterization can be presented in terms of the exponents of the monomials of the self-reciprocal polynomial