… In this note, we will study the growth of gonality in Galois, unramified … Galois, unramified
map between two l-gonal curves over k. Suppose Y has a k-point. If Y does not have a gonal …

… field case, we can prove a stronger result, one which bounds the index of the image of Galois
… Enlarging L cannot increase γL(X), so we may assume L/k is Galois. Choose f ∈ L(X) of …

… Then all cycles of g^ have length 2, so we obtain a 4-gonal embedding Qn … We have seen
how bipartite maps give rise to Belyi pairs, and the converse is also true. If (X,/3) is a Belyi pair, …

… crucial case, we derive the esperantist property by an appeal to the remarkable recent results
concerning expansion … of SL2 (see [34]), and further cases are due to Gill and Helfgott [29], …

… ∞, each cycle of length n corresponding to a 2n-gonal face of B. One can obtain a more …
how pairs of permutations give rise to Belyĭ pairs. The universal covering space of Σο= Σ\{0, …

… Every map Jl is covered by a regular map Jl, corresponding to the core (intersection of all …
this the minimal regular cover of Jl, since it is covered by every regular map which covers Ji. …

… Modular methods typically produce curves whose genus and gonality grow together, which
… 124] to obtain a curve X/Fq and a Galois covering X → P1 with group G whose branch locus …

… (In this section, we take gonality to be a geometric invariant of the curve; that is, in the …
expansion, and if we believe by analogy with the complex case that uniform expansion of Galois …

… We will call gonality of X the minimal gonality of curves in X. … a covering family of k-gonal
curves from a single k-gonal … To limit the growth of notation, we will denote the base changed …

… cover f: C → ℙ 1 of degree d, then f is a Galois cover. … the gonality of the curve is d. Because
the proof needs some complicated calculation, for simplicity, we explain it in the special case …