Local-global principles for large rings of algebraic integers

Let be a global field. In other words, is either a number field, i.e. a finite extension of , or a function field of one variable over a finite field. Denote the algebraic (respectively, separable) closure of by (respectively, by ; cf. also Extension of a field). A prime divisor of is an equivalence class of absolute values (cf. also Norm on a field). For each , let be a representative of . Denote the completion of at by . Then is either or ( is metric), or is a finite extension of or a finite extension of ( is ultra-metric).

There is a natural -topology on whose basic -open subsets have the form , for and , . The -topology has compatible extensions to all sets , where is an affine algebraic set over . In each case, is locally compact.

Embed into the algebraic closure of and let . Then is a real (respectively, algebraic) closure of at if (respectively, ), and is a Henselization of at if is ultra-metric (cf. also Henselization of a valued field). In the latter case, the valuation ring of is denoted by . In each case, is uniquely determined up to a -isomorphism.

If is a set of prime divisors of and is an algebraic extension of , then denotes the set of all extensions to of all .

In the sequel, let be a fixed set of prime divisors of which does not contain all prime divisors. For each algebraic extension of and each , let . Thus, if is metric, then is the -unit ball and if is ultra-metric, then is the valuation ring of . Let . If consists of ultra-metric primes only, then is a Dedekind domain (cf. also Dedekind ring). For example, if and consists of all prime numbers, then .

Fix also a finite subset of . Consider the field of totally -adic numbers:

This is the largest Galois extension of in which each totally splits. Let . If , and is empty, then and is the ring of all algebraic integers. The following two theorems, which can be found in [a10] and [a4], are therefore generalizations of Rumely's local-global principle and the density theorem (cf. also Local-global principles for the ring of algebraic integers).

The local-global principle: In the above notation, let . Consider an absolutely irreducible affine variety over . Suppose that for each . Suppose further that for each . Then .

Here, is the Zariski-open subset of consisting of all non-singular points.

The density theorem: Let and be as in the local-global principle. Let be a finite subset of containing . Suppose that for each . For each , let be a non-empty -open subset of . For each , let be a non-empty -open subset of . Then contains a point which lies in for each .

Although the density theorem looks stronger than the local-global principle, one can actually use the weak approximation theorem and deduce the density theorem from the local-global principle.

Both the local-global principle and the density theorem are actually true for fields which are much smaller than . To this end, call a field extension of PAC over if for every dominating separable rational mapping of absolutely irreducible varieties of dimension over , there exists an such that . If is a number field and consists of ultra-metric primes only, [a8], Thm. 1.4; 1.5, imply both the density theorem and the local-global principle for . In the function field case, [a8] must replace by its maximal purely inseparable extension, which is denoted by . Accordingly, the fields in the assumption of the density theorem and the local-global principle must be replaced by . However, using the methods of [a4] and [a5], it is plausible that even in this case one can restore the theorem for .

By Hilbert's Nullstellensatz (cf. also Hilbert theorem), is PAC over . Hence, [a8], Thms.1.4; 1.5, generalize the density theorem and the local-global principle above. Probability theory supplies an abundance of other algebraic extensions of which are PAC over . The measure space in question is the Cartesian product of copies of the absolute Galois group of equipped with the Haar measure. For each , let be the fixed field of in . By [a7], Prop. 3.1, is PAC over for almost all . Together with the preceding paragraph, this yields the following result (the Jarden–Razon theorem): For every positive integer and for almost all , the field satisfies the conclusions of the local-global principle and the density theorem.

The local-global principle for rings implies a local-global principle for fields. An algebraic extension of is said to be PC (pseudo -adically closed) if each absolutely irreducible variety over which has a simple -rational point for each , has an -rational point. In particular, by the local-global principle and the Jarden–Razon theorem, the fields and are PC for almost all . The main result of [a5] supplies PC extensions of which are even smaller than the fields (the Geyer–Jarden theorem): For every positive integer and for almost all , the field is PC.

Here, is the maximal Galois extension of that is contained in . It is not known (1998) whether satisfies the local-global principle. (So, the Geyer–Jarden theorem is not a consequence of the Jarden–Razon theorem.) Since a separable algebraic extension of a PC field is PC [a9], Lemma 7.2, the Geyer–Jarden theorem implies that is PC for almost all . Likewise, it reproves that is PC.

A field which is PC is also ample (i.e. if is an absolutely irreducible variety over and , then is Zariski-dense in ). Ample fields, in particular PC fields, have the nice property that the inverse problem of Galois theory over has a positive solution (cf. also Galois theory, inverse problem of). That is, for every finite group there exists a Galois extension of such that . Indeed, every finite split embedding problem over is solvable [a11], Main Thm. A, [a6], Thm. 2.

Another interesting consequence of the local-global principle describes the absolute Galois group of : It is due to F. Pop [a11], Thm. 3, and may be considered as a local-global principle for the absolute Galois group of (Pop's theorem): The absolute Galois group of is the free pro-finite product

where is the set of all extensions to of all . This means that if is a finite group, then each continuous mapping whose restriction to each is a homomorphism, can be uniquely extended to a homomorphism .

As a consequence of the local-global principle, Yu.L. Ershov [a2], Thm. 3, has proved that the elementary theory of is decidable. If does not contain , this implies, by [a1], p. 86; Corol. 10, that the elementary theory of is decidable. In particular, Hilbert's tenth problem has an affirmative solution over . If however, , then the elementary theory of is decidable [a3] but the elementary theory of is undecidable [a12].

References

[a1]	L. Darnière, "Étude modèle-théorique d'anneaus satisfaisant un principe de Hasse non singulier" PhD Thesis (1998)
[a2]	Yu.L. Ershov, "Nice local-global fields I" Algebra and Logic , 35 (1996) pp. 229–235
[a3]	M.D. Fried, D. Haran, H. Völklein, "Real hilbertianity and the field of totally real numbers" Contemp. Math. , 74 (1994) pp. 1–34
[a4]	B. Green, F. Pop, P. Roquette, "On Rumely's local-global principle" Jahresber. Deutsch. Math. Ver. , 97 (1995) pp. 43–74
[a5]	W.-D. Geyer, M. Jarden, "PSC Galois extensions of Hilbertian fields" Manuscript Tel Aviv (1998)
[a6]	D. Haran, M. Jarden, "Regular split embedding problems over function fields of one variable over ample fields" J. Algebra , 208 (1998) pp. 147–164
[a7]	M. Jarden, A. Razon, "Pseudo algebraically closed fields over rings" Israel J. Math. , 86 (1994) pp. 25–59
[a8]	M. Jarden, A. Razon, "Rumely's local global principle for algebraic PC fields over rings" Trans. Amer. Math. Soc. , 350 (1998) pp. 55–85
[a9]	M. Jarden, "Algebraic realization of -adically projective groups" Compositio Math. , 79 (1991) pp. 21–62
[a10]	L. Moret-Bailly, "Groupes de Picard et problèmes de Skolem II" Ann. Sci. Ecole Norm. Sup. , 22 (1989) pp. 181–194
[a11]	F. Pop, "Embedding problems over large fields" Ann. of Math. , 144 (1996) pp. 1–34
[a12]	J. Robinson, "On the decision problem for algebraic rings" , Studies Math. Anal. Rel. Topics , Stanford Univ. Press (1962) pp. 297–304.