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 Kinf 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  and in the context of existential definability to Denef . Following Denef , 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 ). 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 ) 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 Kinf, 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 pn. 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 y2 = x3 + 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 (xn, yn), 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(xkm), where d(xkm) is the denominator of the divisor of xkm in the integral divisor semigroup of 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 (xlm) as above refers to the denominator of the divisor of xlm and n xm - k2 refers to the numerator of the divisor
of ^ — k2.
Given u e Kinf integral at some fixed K -prime pK, we now consider a statement of the following sort: Vz e Kinf there exists x, y, x, y e Kinf such that (x, y), (x, y)
1 2 x 2
satisfy the chosen elliptic curve equation and both — and x (u - -) are integral
at pK implying that ^ is integral at pK.
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 Kinf and thus obtain a first-order definition of Z over Kinf. Finally, being able to define Z implies undecidability of the first-order theory of the field.