Polynomial Belyi maps #
Let k be a field and q ∈ k[X] a polynomial with q′ ≠ 0. Regarded as a map of the
projective line, x ↦ q(x) is branched only over ∞ and over the critical values of q, the
values q(ξ) at the roots ξ of q′ (Polynomial.criticalValues). This file proves that
containment in the language of function fields: if F = k(x) over k(t), where t = q(x),
then every place of k(t) at which t is regular and r(t) is a unit, for some
polynomial r vanishing at every critical value of q, is unramified in F
(TauCeti.Place.ramificationIdx_eq_one_of_aeval_eq).
The proof is the derivative criterion for the different (Stichtenoth, Theorem 3.5.10(a)): x is
a root of the monic polynomial (q(X) - t) / lc(q), whose coefficients are regular at such a
place P, so a place P' of F over P is unramified as soon as q′(x) is a unit at P'.
If it is not, then evaluating at P' makes the residue x(P') a root of q′ in the residue
field, so t(P') = q(x(P')) is a critical value of q there, and r(t) vanishes at P.
In characteristic zero, taking r = X (X - 1) gives the criterion for Belyi maps: if the critical
values of q, in every field extension of k, lie in {0, 1}, then F = k(x) over k(t),
t = q(x), is an algebraic Belyi pair (TauCeti.IsAlgebraicBelyiPair.of_criticalValues_subset).
This applies to the Belyi polynomials B_{m,n}
(TauCeti.IsAlgebraicBelyiPair.of_aeval_belyiPolynomial) and, over ℚ, to the polynomials
produced by polynomial Belyification (TauCeti.exists_polynomial_eval_mem_criticalValues_subset).
In Belyi's theorem it is the ramification computation for the last step, which composes a map
t₀ with such a polynomial q: the extension k(t₀) / k(q(t₀)) is unramified away from 0, 1
and ∞.
Main results #
TauCeti.Place.ramificationIdx_eq_one_of_aeval_eq: the places ofk(t)away from∞and from the critical values ofqare unramified ink(x).TauCeti.IsAlgebraicBelyiPair.of_criticalValues_subset: a polynomial whose critical values lie in{0, 1}is an algebraic Belyi pair.TauCeti.IsAlgebraicBelyiPair.of_aeval_belyiPolynomial: the Belyi polynomials are algebraic Belyi pairs.
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Theorem 3.5.10.
- Bernhard Köck, Belyi's theorem revisited, Beiträge zur Algebra und Geometrie 45 (2004), 253–265, §3.
A polynomial map is unramified away from ∞ and its critical values: let F = k(x) over
k(t), where t = q(x) for a polynomial q with q′ ≠ 0, so that F is finite and separable
over k(t). If r ∈ k[X] vanishes
at every critical value of q, in every field extension of k, then every place of F over a
place of k(t) other than ∞ at which r(t) is a unit has ramification index 1.
Polynomials with critical values in {0, 1} are Belyi maps: if F = k(x) over k(t),
where t = q(x) for a polynomial q whose critical values in every field extension of k lie in
{0, 1}, then F is an algebraic Belyi pair.
The Belyi polynomials are Belyi maps: if F = k(x) over k(t), where
t = B_{m,n}(x) for the Belyi polynomial B_{m,n}, then F is an algebraic Belyi pair.