# Defining Z Using Elliptic Curves with Finitely Generated Groups over the Given Field and One Completely q-Bounded Prime

We now return to some ideas we used over number fields: using elliptic curves and the weak vertical method. Below we give an informal description of a construction of a definition of a number field *K* over an infinite algebraic extension K_{inf} of Q using an elliptic curve with a Mordell-Weil group generated by points defined over *K*. This construction also requires one completely *q*-bounded prime *p* (which may equal to *q*). Observe that once we have a definition of *K*, a (first-order) definition of Z follows from a result of J. Robinson.

The use of elliptic curves in the context of definability over infinite extensions also has a long history, as long as the one for norm equations and quadratic forms. Perhaps the first mention of elliptic curves in the context of the first-order definability belongs to Robinson [42] and in the context of existential definability to Denef [7]. Following Denef [8], as has been mentioned above, the author also considered the situations where elliptic curves had finite rank in infinite extensions and showed that when this happens in a totally real field one can existentially define Z over the ring of integers of this field and the ring of integers of any extension of degree 2 of such a field (see [59]). C. Videla also used finitely generated elliptic curves to produce undecidability results. His approach, as discussed above, was based on an elaboration by C.W. Henson of a proposition of J. Robinson and results of D. Rohrlich (see [44]) concerning finitely generated elliptic curves in infinite algebraic extensions.

The main idea behind our construction can be described as follows. Given an element *x e* K_{inf}, we write down a statement saying that *x* is integral at *p* and for every *n e* Z_{>0} we have that *x* equivalent to some element of *K* mod *p*^{n}. By the weak vertical method, this is enough to “push” *x* into *K*. Our elliptic curve as above is the source of elements of *K*. Any solution to an affine equation *y*^{2} = *x*^{3} + *ax + b *of our elliptic curve must by assumption be in *K*. Further if we let *P* be a point of infinite order and let the affine coordinates of [n] *P* corresponding to our equation be *(x _{n}, y*

_{n}), then we remind the reader that the following statements are true:

- 1. Let A be any integral divisor of
*K*and let*m*be a positive integer. Then there exists*k e*Z>_{0}such that A*d(x*where_{km}),*d(x*is the denominator of the divisor of_{km})*x*in the integral divisor semigroup of_{km}*K*. - 2. There exists a positive integer
*m*such that for any positive integers*k, l*,

in the integral divisor semigroup of *K*. Here d *(x _{lm})* as above refers to the denominator of the divisor of

*x*and n

_{lm}*k*

^{xm}-^{2}refers to the numerator of the divisor

^{x}klm

of *^ —* k^{2}.

^{x}klm

Given *u e* K_{inf} integral at some fixed *K* -prime p_{K}, we now consider a statement of the following sort: *Vz* e K_{inf} there exists *x, y, x, y e* K_{inf} such that *(x, y), (x, y)*

1 _{2} *x* _{2}

satisfy the chosen elliptic curve equation and both — and *x (u* - -*)* are integral

*zx x*

*(u ^{2}*_

*)*

^{x}^{2}

at p_{K} implying that ^ is integral at p_{K}.

If *u* satisfies this formula, then since | e *K*, by the weak vertical method we have that *u e K*. Further, if *u* is a square of an integer, this formula can be satisfied. Thus we can proceed to make sure our definition includes all integers, followed by a definition including all rational numbers as ratios of integers, and eventually all of *K* through a basis of *K* over Q. Consequently, at the end of this process we obtain a first-order definition of *K* over K_{inf} and thus obtain a first-order definition of Z over *K*_{inf}. Finally, being able to define Z implies undecidability of the first-order theory of the field.