Positive Stable Rank Condition and Elliptic Curves
We now come back to the discussion of elliptic curves, curves defined by equation y2 = x3 + ax + b, but now with a, b possibly being algebraic integers while A = -16(4a3 + 27b2) is still not equal to 0. We will be looking for solutions to this equation in a specific number field K and will use them in a very different fashion to define integers compared to what we were doing over big rings to define a model of Z. The idea to use elliptic curves with the weak vertical method, as the idea to use an elliptic curve for a model of Z over big rings, belongs to B. Poonen (see ).
The use of the weak vertical method is based on the following properties of points on elliptic curves. 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 the following statements are true over any number field K:
- 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.
It is not hard to understand why the first assertion is true. The reasons are, in some sense, the same as for the assertion that a number field unit e raised to a sufficiently high power is equivalent to 1 modulo any number field divisor. In both cases the reason is the finiteness of residue fields of primes.
Let P be a point of infinite order such that a prime p of a number field K over which the curve is defined, does not occur in the denominators of the affine coordinates of P from some fixed Weierstrass equation of the elliptic curve. (We remind the reader that we can assume this equation is of the form y2 = x3 + ax + b, where a, b are integers of our number field. Thus, x and y have negative order at the same set of primes.)
We now consider our Weierstrass equation over the residue field of p and for the sake of simplicity we will also assume that p does not divide the discriminant of the original equation so that modp we are still looking at an elliptic curve. Given our assumption on the discriminant, P is mapped onto a non-zero element of the group of elliptic curve points. Since the field is finite, the group of points is finite and thus the image of P has a finite order r. Hence [r ] P is mapped to a point at infinity of the elliptic curve mod p. Therefore, [r]P must have coordinates with negative order at p. Once p makes it into the denominator, it will persist in all multiples of [r] P. Further, let p be the rational prime below p (or the rational prime p such that (p) OK c p OK) and observe that properties of formal groups imply that [pr]P will have a higher power of p in the denominator of its coordinates. So any divisor of K will divide some multiple of P. This accounts for the first assertion above. The second assertion is a bit harder and also follows from properties of formal groups. A formal proof of both assertions can be found in .
Existence of a point of infinite order implies that the Mordell-Weil group, the group of points on the elliptic curve in question, is of positive rank. We will always have this assumption when discussing the use of elliptic curves for our definitional purposes. Unfortunately the properties above by themselves are not enough to make elliptic curves usable with the weak vertical method. We also need a stable rank assumption. We want the rank of Mordell-Weil group unchanged whether we look at points with coordinates in K or points with coordinates in some subfield L below. If the rank is unchanged then a fixed integer multiple of any point on the curve has its coordinates derived from our equation in L.
Assume for the purpose of simplification that every point on the curve has its coordinates in the field below and that the class number of K is 1, or in other words given an integral divisor A = p1... pK we can find an integer x such that n(x) = A and therefore we can write any y e K as a ratio of two integers x1 and x2 with n(x1) being relatively prime to n(x2). Then we can write the x-coordinate of every point on the elliptic curve as a ratio of two algebraic integers which are relatively prime.
With these assumptions, we can now consider the following system of equations:
Here (f-, Ц-), i = 1, 2in (3.7) represent two points on our elliptic curve with coordinates written as ratios of integers. Equation (3.8) is the same as Eq. (3.6) but rewritten in terms of our variables taking integer values only. Finally (3.9) is the height bound equation, where m, C are positive integers depending on K only.
If for some element t e OK we can find values for и 1, v1, u2, v2, a1, b1, a2, b2, u, w e OK, then we can deduce from (3.7)-(3.9) that t2 - z = 0 mod b2 in OK, where z e OL and b2 is of much larger height than t. Thus, by the weak vertical method we conclude that t e L. Note that we cannot conclude that z e Z. This would only follow if we also knew that our elliptic curve had rank equal to one and we would need additional equations. However, we do know that if t is an integer, than by arranging (a, Ц-) to be multiples of the same infinite order point, as described above, we can find the values for other variables to satisfy the equations. Thus, applying the weak vertical method, we end up defining a subset of OL containing Z. This is actually enough to define OL, because we can continue to define via ratios a subset of L containing Q and then using a basis of L over Q all elements of OL.
The discussion above leaves us with two questions: how to get down to Z and when do we have a stable rank situation in the first place. We answer the second question first via a Theorem proved by B. Mazur and K. Rubin (see ).
Theorem 3.8 Suppose K/L is a cyclic extension of prime degree of number fields. If the Shafarevich-Tate Conjecture is true for L, then there is an elliptic curve E over L with rank(E(L)) = rank(E(K)) = 1.
Combining this theorem with the weak vertical method, we get an immediate corollary.
Corollary 3.4 Suppose K /L is a cyclic extension of prime degree of number fields and the Shafarevich-Tate Conjecture is true for L. In this case OL has a Diophantine definition over OK .
Returning to the first question we asked above about getting down to Z, we are now in position to note that conditional on Shafarevich-Tate conjecture holding for all number fields, Corollary 3.4 implies that Z is existentially definable over OL for all number fields L. Connecting the cyclic cases to an arbitrary extension L of Q takes several steps:
- 1. Let M be the Galois closure of L over Q. In this case if Z has a Diophantine definition over OM, then Z has a Diophantine definition over OL. Thus without loss of generality, we can assume that L is Galois over Q. The fact that we can always replace a given field by its finite extension, follows from the fact that any polynomial equation with variables ranging in a finite extension can be rewritten as an equivalent polynomial equation with variables ranging in a given field.
- 2. Let L/Q be a Galois extension of number fields. Let KiKn be all the cyclic subextensions of L, i.e. all the subfields Ki of L such that L/Ki is cyclic. Observe that there are only finitely many such subextensions,
and therefore if each OKi has a Diophantine definition over OL, then Z has a Diophantine definition over OL. (Thus, it is enough to show that in every cyclic extension the ring of integers below has a Diophantine definition over the ring of integers above.)
3. If L c H c M is a finite extension of number fields, OH has a Diophantine definition over OM, and OL has a Diophantine definition over OH, then OL has a Diophantine definition over OM. Thus, it is enough to consider cyclic extensions of prime degree only. The reason why this reduction works are pretty transparent. For example suppose PH (t, x) with x = (x1,...,xr) is a Diophantine definition of OL over OH and let PM (t, y) be a Diophantine definition of OH over OM. Now consider the system
Now it is not hard to see that this system has solutions over OM if and only if x1, ...,xr e OH and t e OL. For a general discussion of reductions of this sort see  and Chap. 2 of .
The results above represent the state of our knowledge concerning the status of HTP over the rings of integers of number fields. We also know quite a few things about big subrings of number fields. One could say that the big ring problem is simultaneously easier and harder when considered over extensions. In the next section we start with the easier part.