It provides a necessary condition ((*) below) on branch points z₁,...,z_r and inertia conjugacy classes C₁,...,C_r for a cover f : X → P¹_z (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 Q^cyc, and its intersection with K by K_cyc. Assume for simplicity f is a G-Galois cover, geometrically Galois, with group G. The inertia groups I above z_i are  conjugate cyclic subgroups of G, of order e_i. Over the complexes we can normalize to regard I as generated by some g_i in G, interpreted as follows. There are embedding of the functions on X into Puiseux expansions in  (z-z_i)^1/e_i. Regard g_i as induced by restriction of (z-z_i)^1/e_i → e^2πi/e_i(z-z_i)^1/e_i to these functions. 

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

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

The class C_i is the conjugacy class in G of g_i. Choosing generators of the inertia groups  is compatible with the absolute Galois group G_K=Gal(Ǩ/K) acting on the collection {e^2πi/e}^_e=2∞ via the cyclotomic character ß.

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

 (*)  {(z₁,C₁),...,(z_r,C_r)} = {(z₁^µ,C₁^1/β(µ)}),...,(z_r^µ,C_r^1/β(µ))} (for all µ in G_K). Here ß(µ) is the value of µ restricted to roots of 1, and interpreted as a profinite invertible integer.

So, {C₁,...,C_r} is K-rational: {C₁ⁿ,...,C_rⁿ}={C₁,...,C_r} for any n prime to each e_i and trivial on K_cyc. Recall the notion of Frattini cover ψ: H → G 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.

1. Realizing Z/p^k+1 regularly over Q(T) requires at least φ(p^k+1) branch points (φ is the Euler function). In particular,  Z_p (the p-adic numbers) cannot be regularly realized over Q(T).
2. Suppose a given class C_i is K-rational (i.e. C_iⁿ=C_i for each n prime to e_i and trivial in its action on K_cyc), and is different from other classes. Then, the corresponding z_i  is K-rational.
3. Generalizing #1, for any group G, consider any collection of distinct p-Frattini covers {H_k}_k=1^∞ of G. For each k, let B_k be the minimal number of branch points of a regular realization of G over a fixed number field K. Then B_k → ∞  as k → ∞  (Thm. 4.4 of fried-kop97.pdf).

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  {ς_e}_e=2^∞ of roots of 1. Then the above still works with ς_e replacing e^2πi/e. In positive characteristic the result works for tamely ramified covers, so you can remove from {ς_e}_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,...,C_r are distinct. Rather, understood this as an r-tuple modulo the permutations of S_r.

Pierre Debes and Mike Fried 12/18/07