The Branch Cycle Lemma

The B(ranch) C(ycle) L(emma) first appeared in M. Fried, The field of definition of function fields and a problem in the reducibility of polynomials in two variables, Ill. J. Math. 17, (1973), 128–146 (to solve Davenport's Problem). Then, in general form in M. Fried, Fields of Definition of Function Fields and Hurwitz Families and; Groups as Galois Groups, Comm. in Alg. 5 (1977), 17–82. Several books apply it to various contexts, notably H. Voelkein, Groups as Galois Groups, Cambridge Studies in Advanced Mathematics 53 (1996), p. 34.

It provides a necessary condition ((*) below) on branch points z1,...,zr and inertia conjugacy classes C1,...,Cr for a cover f : XP1z (the Riemann sphere uniformized by the variable z) to be defined over some subfield K of the complex numbers. This works for tamely ramified covers in positive characteristic, too.

The lemma works for many types of covers – defined by different equivalences. It gives the first invariant, or condition, on a field K for there to be a cover of the given type over K. A brief history of the BCL and the use of Weil's co-cycle condition applied to covers is in [exceptTowYFFTA_519.pdf, App. B.2.2].

Denote the field of all roots of unity over Q by Qcyc, and its intersection with K by Kcyc. Assume for simplicity f is a G-Galois cover, geometrically Galois, with group G. The inertia groups I above zi are  conjugate cyclic subgroups of G, of order ei. Over the complexes we can normalize to regard I as generated by some gi in G, interpreted as follows. There are embedding of the functions on X into Puiseux expansions in  (z-zi)1/ei. Regard gi as induced by restriction of (z-zi)1/ei → e2πi/ei(z-zi)1/ei to these functions.

Riemann's Existence Theorem says we can choose these gis with these properties:

They generate G; and satisfy product-one: g1 ··· gr=1 (in the given order).

The class Ci is the conjugacy class in G of gi. Choosing generators of the inertia groups  is compatible with the absolute Galois group GK=Gal(Ǩ/K) acting on the collection {e2πi/e}e=2 via the cyclotomic character ß.

Statement of the BCL: If f has definition field K, then GK acts on the invariant {(z1,C1),...,(zr,Cr)} trivially. That is:

(*)  {(z1,C1),...,(zr,Cr)} = {(z1µ,C11/β(µ)}),...,(zrµ,Cr1/β(µ))} (for all µ in GK). Here ß(µ) is the value of µ restricted to roots of 1, and interpreted as a profinite invertible integer.

So, {C1,...,Cr} is K-rational: {C1n,...,Crn}={C1,...,Cr} for any n prime to each ei and trivial on Kcyc. Recall the notion of Frattini cover ψ: HG of a group G: It is a covering homomorphism, and no proper subgroup of H maps surjectively to G. The rank of H – minimal number of generators – is the same as the rank of G.

Then, ψ is p-Frattini if the kernel is a p group. As p-Frattini-cov.html recounts, the kernel of such a cover is a nilpotent group whose p-Sylows are nontrivial only for the primes dividing |G|.  Also, for each such prime, there are infinitely many distinct such p-Frattini covers.

Consequences of the BCL:
1. Realizing Z/pk+1 regularly over Q(T) requires at least φ(pk+1) branch points (φ is the Euler function). In particular,  Zp (the p-adic numbers) cannot be regularly realized over Q(T).
2. Suppose a given class Ci is K-rational (i.e. Cin=Ci for each n prime to ei and trivial in its action on Kcyc), and is different from other classes. Then, the corresponding zi  is K-rational.
3. Generalizing #1, for any group G, consider any collection of distinct p-Frattini covers {Hk}k=1 of G. For each k, let Bk be the minimal number of branch points of a regular realization of G over a fixed number field K. Then Bk → ∞  as k → ∞  (Thm. 4.4 of fried-kop97.pdf).
The 1995 paper Extension of Constants, Rigidity, and the Chowla-Zassenhaus Conjecture has many elementary examples of its use, as does the 1995 Enhanced review of J.-P. Serre's Topics in Galois Theory, with examples illustrating braid rigidity. A more sophisticated consequence appears in mt-overview.html under "How MTs Arise." A history of the BCL is in [exceptTowYFFTA_519.pdf, App. B1]

Adjustments for no given explicit roots of 1, or for positive characteristic:

If no embedding of K in the complexes is given, stating the lemma precisely requires picking a coherent system  {ςe}e=2 of roots of 1. Then the above still works with ςe replacing e2πi/e. In positive characteristic the result works for tamely ramified covers, so you can remove from {ςe}e=2 all values of e divisible by the characteristic. For wildly ramified covers there is a whole different rubric (fr-mez.pdf), though it includes the tamely ramified case.

We don't assume the classes C1,...,Cr are distinct. Rather, understood this as an r-tuple modulo the permutations of Sr.

Pierre Debes and Mike Fried 12/18/07