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 ...

We discuss sufficient conditions for a given curve to be covered by a maximal curve with the covering being unramified; it turns out that the given curve itself will be also maximal. We relate our main result to the question of whether or not a maximal curve is covered by the Hermitian curve. We also provide examples illustrating the results. §1. Let X be a projective, geometrically irreducible...

In a previous paper ([10]), the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and Q( √ −3). Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points o...

In [11], a new bound for the number of points on an algebraic curve over a "nite "eld of odd order was obtained, and applied to improve previous bounds on the size of a complete arc not contained in a conic. Here, a similar approach is used to show that a complete arc in a plane of even order q has size q#2 or q!Jq#1 or less than q!2Jq#6. To obtain this result, "rst a new characterization of a ...

This paper presents a method to smoothly interpolate a given sequence of solid orientations using circular blending quaternion curves. Given three solid orientations, a circular quaternion curve is constructed that interpolates the three orientations. Therefore, given four orientations q i?1 ; q i ; q i+1 ; q i+2 , there are two circular quaternion curves C i and C i+1 which interpolate the tri...

A sequence of rational functions in a variable q is q-holonomic if it satisfies a linear recursion with coefficients polynomials in q and qn. In the paper, we assign a tropical curve to every q-holonomic sequence, which is closely related to the degree of the sequence with respect to q. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the kn...

We attempt to determine which classes of algebraic varieties over Q must have points in some abelian extension of Q. We give: (i) for every odd d > 1, an explicit family of degree d, dimension d − 2 diagonal hypersurfaces without Qab-points, (ii) for every number field K, a genus one curve C/Q with no K ab-points, and (iii) for every g ≥ 4 an algebraic curve C/Q of genus g with no Qab-points. I...