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Saltman’s Generic Galois Extensions and Problems in Field Theory

David Harbater May 16, 2011

Motivation from Inverse Galois Problem

Saltman, Advances in Math. (1982) Motivated by (inverse) Galois theory; has motivated further work. IGP: Is every finite group G a Galois group over Q? Strategy: Realize G over Q(x); specialize x to α ∈ Q. Example: G = C2. Adjoin y: y2 = x to Q(x). Can specialize x to 2; get Q[ √ 2]/Q.

Can always specialize: Hilbert’s Irreducibility Theorem: If f (x, y) ∈ Q[x, y] is irreducible, then for infinitely many α ∈ Q, f (α, y) ∈ Q[y] is irreducible. Also works with more variables. So: If G is a Galois group over Q(x1, . . . , xn), then G is a Galois group

• ver Q.

Similarly over any global field (number field or function field of a curve over a finite field); such fields are Hilbertian.

This strategy has been used obtain many groups as Galois groups

• ver Q.

1) Use geometry to get a branched cover of P1

C with group G.

2) Choosing rational branch points, the cover is defined over ¯ Q. 3) Sometimes one can show it’s defined over Q. 4) The extension of function fields is Galois over Q(x); also regular (Q is algebraically closed in the extension of Q(x)). Re 3): Rigidity (Matzat, Thompson, Belyi; Fried, Shih); e.g. monster group is a Galois group over Q.

Noether’s problem

An early attempt at IGP/Q via IGP/Q(x1, . . . , xn), due to Emmy Noether: Given G, let G act transitively on {1, . . . , n} (e.g. by regular rep.). Get an action of G on L := Q(y1, . . . , yn); invariants K = LG. Question: Is K ∼ = Q(x1, . . . , xn)? If so, Hilbert Irreducibility ⇒ G is a Galois group over Q.

• Example. G = Sn acting on {1, . . . , n}. Invariants in Q(y1, . . . , yn):

Q(σ1, . . . , σn). So yes.

Geometrically: Q(y1, . . . , yn) is the function field of An

Q, with an

action of G. Take the quotient An

Q/G. This is unirational. Is it

rational? Over an algebraically closed field, it’s hard to find examples of non-rational unirational varieties. There are more examples over non-algebraically closed fields (twists).

Noether’s problem: Determine if Q(y1, . . . , yn)G is purely

• transcendental. (If so, G is a Galois group over Q.)

Always? No: Swan 1969, Voskresenskii 1970: no if G = C47, C113, C223, . . . Lenstra 1974: no if G = C8, or for abelian groups containing C8.

Over an algebraically closed field, is the answer to Noether’s problem always yes? Answer: No (Saltman, 1984). If k is algebraically closed, and p = char(k), then there is a p-group G of order p9 such that the fixed field is not purely transcendental. This gives new examples of varieties that are unirational but not rational, over an algebraically closed field. Later: related examples by Bogomolov (1987), Plans (2007). The proofs use unramified cohomology (in Saltman’s case, the unramified Brauer group).

Generic Galois extensions

Notion due to Saltman, related to Noether’s problem. Ask for existence of such an extension for a given group G. The answer is yes more often than for Noether’s problem. Yes ⇒ G is a regular Galois group over Q(x) and so G is a Galois group over Q. Existence of such an extension gives a structural result about all G-Galois extensions of Q, and over all fields containing Q. (As in the earlier discussion, can work over other fields as well.) Generic Galois extensions are related to moduli spaces, but allowing repetition.

The definition uses the notion of a Galois extension of rings. Recall: if a finite group G acts on a field L, then a field extension L/K is G-Galois ⇔ K = LG and L ⊗K L ∼ =

g∈G L.

Analogously, if G acts on a ring S, then we say that a ring extension S/R is G-Galois if R = SG and S ⊗R S ∼ =

g∈G S.

• Example. G = C2, R = Q[x, x−1], S = R[y]/(y2 − x).

Note: The definition rules out ramification. Can’t take R′ = Q[x] and S′ = R′[y]/(y2 − x). Geometrically: Spec(S) → Spec(R) is a G-Galois étale cover.

A generic Galois extension for G over a field k is a G-Galois ring extension S/R = k[x1, . . . , xn, 1/g] such that ∀ field k′ ⊇ k, ∀ G-Galois extension ℓ′/k′ (not necessarily a field), ℓ′/k′ is a specialization of S/R (by setting xi = αi ∈ k′). Geometrically: Spec(R) = U ⊆ An

k, a Zariski open dense subset.

We have a G-Galois étale cover of U, corresponding to a G-Galois branched cover of An

k with ramification away from U.

This cover has the property that it parametrizes all G-Galois extensions of fields k′ ⊇ k, by taking the fibers over k′-points of U. A strong condition; does such an extension exist?

Saltman: If Noether’s problem has a positive answer for G, then there is a generic Galois extension for G over Q. So in particular there is a generic Galois extension for Sn. But existence of a generic Galois extension is more general than a positive answer to Noether’s problem: Saltman: If n is odd, then there is a generic Galois extension for Cn. (More generally, if n = 2rs, with s odd, then there is a generic Galois extension for Cn over k iff k(ζ2r )/k is cyclic.) In particular, ∃ generic Galois extension for C47 over Q, even though the answer to Noether’s problem is no.

How to construct such extensions? Simple case: G = Cn, and work over k ∋ ζn. By Kummer theory, every Cn-Galois extension of k is of the form k[y]/(yn − α), with α ∈ k. So a generic Galois extension is given by R = k[x, x−1], S = R[y]/(yn − x). What if we don’t have ζn? How to get a generic Galois extension for Cn?

Example: C3 over Q. A C3-Galois extension of Q is given by Q(ζ7 + ζ−1

7 ). But this is just one extension. How to get a family?

We want an extension of Q[x, 1/g] that specializes to various (even all!) C3-Galois extensions of Q. Idea: Build a non-constant C3-Galois extension L′/K ′ = Q(ζ3, x) by Kummer theory, such that it descends to a C3-Galois extension L/Q(x).

To do this, the action of Γ = Gal(Q(ζ3)/Q) should lift to an action

• n L′ that commutes with the action of C3 on L′.

So L′/Q(x) is Galois with group C3 × Γ. Then take L = (L′)Γ. L′

Γ C3

L

C3

K ′

Γ

Q(x)

Explicitly: Take x = (x1, x2). Over K ′ = Q(ζ3, x1, x2), take L′ = K ′[y]/(y3 − s), where s = x1 + ζ3x2 x1 + ζ−1

3 x2

∈ K ′. Γ = Gal(Q(ζ3)/Q) = Gal(Q(ζ3, x1, x2))/Q(x1, x2)) The generator of Γ takes ζ3 to ζ−1

3 , and s to s−1.

This extends to an action on L′ by y → y−1, commuting with the action of C3 on L′. u := s + s−1 is invariant under Γ and so lies in Q(x1, x2); in fact u = (2x1 − x2 − 2x1x2)/(x2

1 − x1x2 + x2 2).

v := y + y−1 is also invariant under Γ, but not under C3; so v ∈ K ′. So v generates L := (L′)Γ over Q(x1, x2). Its minimal polynomial over Q(x1, x2) is V 3 − 3V − u. This defines a C3-Galois regular extension of Q(x1, x2).

This C3-Galois extension of Q(x1, x2) gives a generic extension for C3 over Q, by taking the integral closure of Q[x1, x2] in the extension and inverting the discriminant. Saltman shows how to do this more generally for Cq with q an odd prime power (using more xi’s). Taking products, can get Cn for all odd n. (Can replace Q by other fields) For C2r this can still be done if adjoining ζ2r gives a cyclic extension. So we get a generic Galois extension for an abelian group A over k if k(ζ2r )/k is cyclic, where 2r is the highest power of 2 dividing the exponent of A.

We can also get generic Galois extensions for many semi-direct

• products. For example, Saltman showed:

1) If there are generic Galois extensions for N and H over k then there is a generic Galois extension for N ≀ H over k. 2) If A, H have relatively prime order, with A abelian, and if there are generic Galois extensions for A and H over k, then there is a generic Galois extension for A ⋊ H over k. 3) If q is an odd prime power and ζq ∈ k, and if ∃ a generic Galois extension for H over k, then ∃ a generic Galois extension for Cq ⋊ H over k. (Also for a 2-power q if the image of H in Aut(Cq) is cyclic.) In particular, ∃ a generic Galois extension for Dq over k if ζq ∈ k.

Related results by others: Elena Black, 1999: If q is a power of 2 then there is a Dq generic Galois extension over k of char = 2 if q = 4 or 8, or ζq + ζ−1

q

∈ k. Mestre, Rikuna, 2006: ∃ a generic Galois extension for ˜ A4 over Q. Plans, 2007: ∃ a generic Galois extension for ˜ A5 over Q. Serre, 2003: There is no generic Galois extension for ˜ A6, ˜ A7 over Q. Plans, 2007: ∃ a generic Galois extension for ˜ S4 = GL(2, 3) in char = 2; in fact, there is a positive answer to Noether’s problem. Jensen-Ledet-Yui, Generic Polynomials, 2002: Explicit expression for a generic extension for Q8 if char = 2 (also many other groups).

Generic Galois extensions and IGP

Given a group G, primes p1, . . . , pn, and G-Galois extensions Ai/Qpi (not necessarily fields), is there a G-Galois extension A/Q such that A ⊗Q Qpi ∼ = Ai for all i? Informally: “Can we patch local Galois extensions, to obtain extensions of Q?” If so, suppose G is generated by G1, . . . , Gn ⊆ G (e.g. cyclic subgroups), and that we have Gi-Galois field extensions Li/Qpi. Let Ai =

G/Gi Li = IndG GiLi, a G-Galois algebra over Qpi. By

patching we get a G-Galois field extension of Q. If this could work in general, it would give a solution to the inverse Galois problem over Q.

A weak form was stated as a theorem by Grunwald, 1933: It said: if G is cyclic, then the answer to the question is yes. This was used at the time in connection with the Brauer-Hasse-Noether Theorem: BHN proved a local-global principle for Brauer groups: If a central simple algebra A over a number field k splits over every completion

• f k, then it splits over k.

Combined with Grunwald’s assertion, it gave: every central simple algebra over a number field is cyclic. Grunwald’s assertion also implied a local-global principle: If a ∈ k is an n-th power in almost every completion, then it is an n-th power.

In 1942, Wang found a counterexample to Grunwald’s assertion: There is no C8-Galois extension A/Q such that A ⊗Q Q2 is the unramified C8-extension field of Q2. (Also, 16 is an 8-th power in Qp for all odd p, but not in Q, nor in

• Q2. This contradicts the local-global principle.)

Wang also showed that all counterexamples are “essentially” of this form and relate to the prime 2 (having to do with Q(ζ8)/Q not being cyclic). The modified assertion (Grunwald-Wang theorem) was still sufficient to prove the theorem on cyclic algebras.

As for “patching field extensions” in the number field context: Saltman: If there is a generic Galois extension for G over k, and if Ai/ki (for i = 1, . . . , n) are G-Galois extensions, then there is a G-Galois extension A/k such that A ⊗k ki ∼ = Ai. So in this situation we can find a G-Galois extension with specified local behavior (a strong IGP). (Of course in this case we already knew that G was a Galois group

• ver k.)

Combined with Wang’s counterexample for G = C8, we get another proof that there is no generic Galois extension for C8 over Q; and an explanation of what the obstruction is.

Arithmetic lifting problem

A related notion first studied by Sybilla Beckmann and then Elena

• Black. (Beckmann-Black problem)

For a general group G, does every G-Galois extension of Q arise as the specialization of a regular Galois extension of Q(x)? (I.e. from a branched cover of the line.) Posed by Sybilla Beckmann (1994), proven if G is abelian or a symmetric group. Elena Black (1999) proved this for more groups, and in particular: If there is a generic Galois extension for G over k, then arithmetic lifting holds for G over k. Arithmetic lifting is more general than generic Galois: arithmetic lifting holds for C8 over Q (since it’s abelian) but there is no generic Galois extension for C8 over Q.

Noether Problem

• Generic Galois
• Patching
• Arith. Lifting

There are no known examples where arithmetic lifting fails. Beckmann-Black conjecture: Every finite group satisfies arithmetic lifting over every field.

BB is an “inverse IGP”: if yes, every Galois realization arises geometrically, as a specialization of a branched cover of the line. This expectation was a motivation for the Noether problem and generic Galois extensions, and for the main approach to IGP. Another connection to IGP: Dèbes (1999): If the BB conjecture is true, then there is an affirmative answer to RIGP over every field. That is, BB ⇒ every finite group G is the Galois group of a regular extension of k(x) for all k. So if k is Hilbertian (e.g. if k is a global field), then G is a Galois group over k. This is extremely strong (IGP unknown for global fields); so BB would be hard to prove.

Function fields of curves over complete valued fields

In an analogous context, things become easier: function fields F of curves over a complete discretely valued field; e.g. Qp(x). There, a form of patching holds, and so we get IGP for such fields (D.H., 1987). By specialization, this implies IGP for function fields

• ver algebraically closed fields.

It also implies IGP for function fields over large fields (F. Pop, 1996). Moreover, arithmetic lifting holds there, via patching (Colliot-Thélène 2000, Moret-Bailly 2001).

These arguments used either Grothendieck’s formal schemes or Tate’s rigid analytic spaces. A later approach avoided some of this machinery (Haran-Voelklein-Jarden). More recently: a more elementary approach (D.H. and Julia Hartmann), that can be applied more generally. In joint work with Daniel Krashen: used this to obtain local-global principles for Brauer groups (analogous to Brauer-Hasse-Noether), u-invariant, torsors under linear algebraic groups.

In applications to IGP over fields like Qp(x), in order to use patching we need to have cyclic extensions locally. If have ζn, then use Kummer theory. Otherwise, use the explicit descriptions of cyclic extensions given in Saltman’s construction of generic Galois extensions. So in this context: we have patching even without assuming generic Galois extensions; but these extensions in the case of cyclic groups are nevertheless relied on in order to apply patching to solve problems like IGP and ALP. So again they are quite useful.