An elementary proof of the group law for elliptic curves

We give an elementary proof of the group law for elliptic curves using explicit formulas.Comment: 9 pages, 1 figur

Novikov homology and noncommutative Alexander polynomials

In the early 2000's Cochran and Harvey introduced non-commutative Alexander polynomials for 3-manifolds. Their degrees give strong lower bounds on the Thurston norm. In this paper we make the case that the vanishing of a certain Novikov-Sikorav homology module is the correct notion of a monic non-commutative Alexander polynomial. Furthermore we will use the opportunity to give new proofs of several statements about Novikov-Sikorav homology in the three-dimensional context.Comment: 30 pages, to appear in a special volume of JKTR in memory of Tim Cochra

Twisted Reidemeister torsion, the Thurston norm and fibered manifolds

We prove that the twisted Reidemeister torsion of a 3-manifold corresponding to a fibered class is monic and we show that it gives lower bounds on the Thurston norm. The former fixes a flawed proof in [FV10], the latter gives a quick alternative argument for the main theorem of [FK06].Comment: 11 pages, 1 figur

The profinite completion of 33-manifold groups, fiberedness and the Thurston norm

We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the profinite completion of knot groups distinguish knots and show that it distinguishes each torus knot and the figure eight knot among all knots. We show also that it distinguishes between hyperbolic knots with cyclically commensurable complements under the assumption that their Alexander polynomials have at least one zero which is not a root of unity.Comment: 22 page