# 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(4a^{3} + 27b^{2}) = 0. This equation defines an elliptic curve (a non-singular plane curve of genus 1).

All the points *(x, y) e* Q^{2} 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 = (x _{P}, y_{P}), Q = (xq , yQ), R = (x_{R}, y_{R})* 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, Z*

*2*

*, Z*

*3*

*, Z*

*4*

*), g(zu Z*

*2*

*, z*are fixed (somewhat unpleasant looking)

_{3}, z_{4})*rational functions.*Further, -

*P = (x*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

_{P}, -y_{P}).*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 = (x*). (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

_{n}, y_{n}*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 [3]) 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 of*Z *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.