Namespaces
Variants
Actions

Difference between revisions of "Local-global principles for large rings of algebraic integers"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (link)
Line 23: Line 23:
 
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120116.png" /> which are much smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120117.png" />. To this end, call a field extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120118.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120119.png" /> PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120121.png" /> if for every dominating separable rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120122.png" /> of absolutely irreducible varieties of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120123.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120124.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120125.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120126.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120127.png" /> is a number field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120128.png" /> consists of ultra-metric primes only, [[#References|[a8]]], Thm. 1.4; 1.5, imply both the density theorem and the local-global principle for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120129.png" />. In the function field case, [[#References|[a8]]] must replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120130.png" /> by its maximal purely inseparable extension, which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120131.png" />. Accordingly, the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120132.png" /> in the assumption of the density theorem and the local-global principle must be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120133.png" />. However, using the methods of [[#References|[a4]]] and [[#References|[a5]]], it is plausible that even in this case one can restore the theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120134.png" />.
+
Both the local-global principle and the density theorem are actually true for fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120116.png" /> which are much smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120117.png" />. To this end, call a field extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120118.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120119.png" /> PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120121.png" /> if for every dominating separable rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120122.png" /> of absolutely irreducible varieties of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120123.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120124.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120125.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120126.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120127.png" /> is a number field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120128.png" /> consists of ultra-metric primes only, [[#References|[a8]]], Thm. 1.4; 1.5, imply both the density theorem and the local-global principle for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120129.png" />. In the function field case, [[#References|[a8]]] must replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120130.png" /> by its maximal [[purely inseparable extension]], which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120131.png" />. Accordingly, the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120132.png" /> in the assumption of the density theorem and the local-global principle must be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120133.png" />. However, using the methods of [[#References|[a4]]] and [[#References|[a5]]], it is plausible that even in this case one can restore the theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120134.png" />.
  
 
By Hilbert's Nullstellensatz (cf. also [[Hilbert theorem|Hilbert theorem]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120135.png" /> is PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120136.png" />. Hence, [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120137.png" /> which are PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120138.png" />. The measure space in question is the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120139.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120140.png" /> copies of the absolute [[Galois group|Galois group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120141.png" /> equipped with the [[Haar measure|Haar measure]]. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120142.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120143.png" /> be the fixed field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120144.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120145.png" />. By [[#References|[a7]]], Prop. 3.1, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120146.png" /> is PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120147.png" /> for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120148.png" />. Together with the preceding paragraph, this yields the following result (the Jarden–Razon theorem): For every positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120149.png" /> and for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120150.png" />, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120151.png" /> satisfies the conclusions of the local-global principle and the density theorem.
 
By Hilbert's Nullstellensatz (cf. also [[Hilbert theorem|Hilbert theorem]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120135.png" /> is PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120136.png" />. Hence, [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120137.png" /> which are PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120138.png" />. The measure space in question is the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120139.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120140.png" /> copies of the absolute [[Galois group|Galois group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120141.png" /> equipped with the [[Haar measure|Haar measure]]. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120142.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120143.png" /> be the fixed field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120144.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120145.png" />. By [[#References|[a7]]], Prop. 3.1, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120146.png" /> is PAC over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120147.png" /> for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120148.png" />. Together with the preceding paragraph, this yields the following result (the Jarden–Razon theorem): For every positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120149.png" /> and for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120150.png" />, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120120/l120120151.png" /> satisfies the conclusions of the local-global principle and the density theorem.

Revision as of 17:16, 7 November 2016

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.
How to Cite This Entry:
Local-global principles for large rings of algebraic integers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local-global_principles_for_large_rings_of_algebraic_integers&oldid=16794
This article was adapted from an original article by Moshe Jarden (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article