HTP over the Rings of Integers of Number Fields
The state of knowledge concerning the rings of integers and HTP is summarized in the theorem below.
Theorem 3.7 Z is Diophantine and HTP is unsolvable over the rings of integers of the following fields:
- • Extensions of degree 4 of Q (except for a totally complex extension without a degree-two subfield), totally real number fields and their extensions of degree 2. (See [6, 8].) Note that these fields include all abelian extensions.
- • Number fields with exactly one pair of non-real embeddings (See [30, 48].)
- • Any number field K such that there exists an elliptic curve E defined over Q with E (Q) of positive rank with [ E (K) : E (Q)] < to. (See [31, 33, 58].)
• Any number field K such that there exists an elliptic curve E defined over K with E (K) of rank 1, and an abelian variety V defined over Q such that V (Q) and V (K) have the same rank. (See .)
All the results above concerning rings of integers are derived by constructing a Diophantine definition of Z over the rings in question and they all follow what we have called elsewhere a “strong” or “weak vertical method”.
Both methods rely on congruences to “force” an element t from a ring R of integers of a number field K to take values in Z. These congruences are of the form
where n e Z and w e R and of bigger height relative to t .In the strong version of the method the height of w is also large relative to n and this forces the equality t = n to hold.
In the weak version of the method we don’t have a bound on n, but we know that w e Z. In this case, again assuming the height of w is sufficiently large relative to t, we conclude that t e Q but not necessarily equal to n.
One typical way to produce a congruence (3.4) is to isolate powers of a single unit e in the ring of integers. (A unit is an invertible element of the ring.) If one succeeds in doing this, the elementary algebra produces the first ingredient of the congruence.
in the ring of integers of the number field, where k, n e Z>0. In other words we have a divisibility condition
in R. Thus, if we write
then we are in fact writing t = n mod w, with w = ek - 1, and we are part of the way there. Also, writing down equations affirming that the height of t is small relatively to ek - 1 is not that complicated. It can be done through a requirement that some polynomial in t divides ek - 1. It is not hard to show that given any algebraic integer u, there exists k e Z>0 such that u divides ek - 1 in the ring of integers.
Now, if we want to use the strong vertical method, we need to make n small relative to the height of ek - 1. This, unfortunately, is rather hard and requires a pretty intimate knowledge of the equations involved. At the same time, if we want to use the weak vertical method, then we need a way to replace ek - 1 by some w e Z so that w divides ek - 1 and at the same time is still large relative to t.
The weak vertical method can also be used to push t not necessarily all the way down to Z but maybe to a subfleld M of the given field K, so that over M a different method, e.g. the strong one, can be used to complete the descent to Z. If we are using the weak method just to get to a subfield M, we only need w to be in the ring of integers of M. This is often a lot easier, than satisfying the requirement that w is in Z.
The strong vertical method was used by J. Denef over totally real number fields and by T. Pheidas and the author for the fields with one pair of non-real embeddings. (At the time of these results, the method did not have a name.) The construction of a Diophantine definition of Z over the ring of integers for all the other fields listed above used a weak vertical method. The equations used in all constructions were either norm or elliptic curve equations. The last result in Theorem 3.7 also used an abelian variety satisfying a stable rank condition. This condition is discussed in more detail in the next section. Here we would just like to explain briefly the use of norm equations and their limitations with respect to both methods.
The use of norm equations for both vertical methods depends on the interaction of ranks of unit groups in the rings of integers of number fields. First of all, the group of units inside every number field is of finite rank and we have a formula to compute the rank. If K is a number field of degree n over Q with r real embeddings and 2s non-real embeddings into the chosen algebraic closure of Q, then the rank of the unit group is r + s — 1. (Non-real embeddings always come in pairs due to complex conjugation.)
Now if we have a totally real field K and its extension M of degree 2 such that it has exactly two real embeddings, we conclude that the difference in ranks of their unit groups is exactly one. Using the fact that the norm map NM/K : M ^ K maps units to units and is a homomorphism of unit groups of M and K, from the rank calculation we conclude that the kernel of the map, i.e. the set of units whose norm is equal to 1 is a subgroup of the unit group of M of rank 1. (The rank of the kernel is the difference in the M and K unit group ranks.) A finitely generated multiplicative group of rank one is more or less a set of powers of a single element (possibly times elements of finite order, in our case roots of unity.) Writing down a polynomial equation computing the norm using the variables with values in K we can get a polynomial equation whose solutions effectively describe powers of a unit. This equation is quite well known under the name of Pell equation and has a number of convenient properties that we can leverage to bound the heights as described above. Thus we can proceed with the strong vertical method.
The first substantial applications of the weak vertical method (again long before the author of this narrative gave it a name) was due to Denef and Lipshitz  and it was also based on a calculation of the rank of unit groups comprised of solutions to norm equations. We show how the weak vertical method was used in this paper via the field diagram below where we assume that K is a totally real field of degree n over Q, [M : K]= 2, [F : K] = 2, F П M = K, and G is the compositum of M and F (inside the chosen algebraic closure).
Using the formula for computing ranks of unit groups one can choose a field F so that G has no real embeddings while the following equality holds:
Indeed, let rM be the number of real embeddings of M and 2sM the number of nonreal embeddings, so that r2M + sM = n. Let rF, sF be the corresponding numbers for F with r-f + sF = n. Using these notation and our assumptions on G we see that the left side of (3.5) is equal to 2n - rM - sM and the right side is equal to rF + sF - n. Thus we need n - r-f = r-f or 2sM = rF. In other words every embedding of K extended to a real embedding of M, should be extended to a non-real embedding of F and vice versa. Note that this condition on embeddings will also guarantee that all embeddings of G are non-real and M n F = K.
The final piece needed to use the weak vertical method comes from the following observation. Any unit of F with its K-norm equal to one is also a unit of G with the M- norm equal to one. This follows from the fact that M n F = K. Thus, ker NF/K c ker NG/M and ker NF/K is of finite index in ker NG/M since ker NG/M is finitely generated.
Thus, if e e ker NG/M, then for some fixed positive integer n independent of e, it is the case that en is actually an element of ker NF/K c F .Nowlete1 ,e2 e ker NG/M and consider the equation
where r e Z>0 and r = 0 mod n. By the discussion above we can deduce that w e F. So if we have a congruence
with the height of er1 - 1 relatively large to t e OM, then t e OM n F = OK .At the same time, if e2 = ef with m e Z>0, and t = m, then the congruence will hold. Thus we have a foundation for applying the weak vertical method in order to define OK over OM. Once we defined OK, we can continue with the strong vertical method to get all the way down to Z.
Even from this brief description of the way the norm equations are used in the construction of an existential definition of Z, it is clear that this particular use of norm equations can work in special cases only, i.e. when the number fields are totally real or are not “far” from being totally real. So a different foundation for the vertical method is highly desirable. This new foundation is conjecturally provided by elliptic curves.