Elliptic Curves and Diophantine Models
An old plan for building a Diophantine model of Z over Q involved using elliptic curves. Consider an equation of the form:
where a, b e Q and A = -16(4a3 + 27b2) = 0. This equation defines an elliptic curve (a non-singular plane curve of genus 1).
All the points (x, y) e Q2 satisfying (3.2) (if any) together with O —the “point at infinity”—form an abelian group, i.e. there is a way to define addition on the points of an elliptic curve with O serving as the identity. The group law on an elliptic curve can be represented geometrically (see for example [61, Chap. III, Sect. 3]). However, what is important to us is the algebraic representation of the group law. Let P = (xP, yP), Q = (xq , yQ), R = (xR, yR) be the points on an elliptic curve E with rational coordinates. If P +e Q = R and P, Q, R = O, then xr = f (xp, yp, xq, yQ), yR = g(xp, yp, xq, yQ), where f (z, Z2, Z3, Z4), g(zu Z2, z3, z4) are fixed (somewhat unpleasant looking) rational functions. Further, -P = (xP, -yP). Mordell-Weil Theorem (see [61, Chap. III]) tells us that the abelian group formed by points of an elliptic curve over Q is finitely generated, meaning it has a finite rank and a finite torsion subgroup. It is also not very difficult to find elliptic curves whose rank is one. So let E be such an elliptic curve defined over Q such that E(Q) = Z as abelian groups. (In other words E(Q) has no torsion points. In practice torsion points are not an impediment, but they do complicate the discussion.) Let P be a generator and consider a map sending an integer n = 0 to [n] P = (xn, yn). (We should also take care of 0, but we will ignore this issue for the moment.) The group law assures us that under this map the image of the graph of addition is Diophantine. Unfortunately, it is not clear what happens to the image of the graph of multiplication. Nevertheless one might think that we have a starting point at least for our Diophantine model of Z. Unfortunately, it turns out that situation with Diophantine models is not any better than with Diophantine definitions. Further a theorem of Cornelissen and Zahidi (see ) showed that multiplication of indices of elliptic curve points is probably not existentially definable.
Theorem 3.2 If Mazur’s conjecture on topology of rational points holds, then there is no Diophantine model ofZ over Q.
This theorem left HTP over Q seemingly out of reach. It is often the case with difficult Mathematical problems that the search for solutions gives rise to a lot of new and interesting Mathematics, sometimes directly related to the original problem, sometimes only marginally so. People trying to resolve the Diophantine status of Z also proceeded in several directions. The two directions generating the most activity are the the big ring project and attempts to reduce the number of universal quantifiers in first-order definitions of Z over Q. We review the big ring project first.