Belyi's TheoremPosted on December 2, 2018
Belyi’s Theorem is an arithmetic-geometric result that has the unusual quality of being both fairly recent and provable with only the material from a first course in classical algebraic geometry.
In what follows are over unless otherwise specified.
Theorem (Belyi, 1979). Let be an irreducible non-singular projective curve defined as the zero set of Then there exists a finite regular map ramified only over a subset of
The proof strategy takes a projection of onto and then use a sequence of regular maps to gather the ramification points of into the set
Lemma 1. The projection away from the point is a finite regular map ramified over the points such that
Let Then is ramified over if and only if there exists such that Note that is a local parameter for , where is any linear form not vanishing at Thus Hence if and only if is zero on the tangent line The tangent line is described by We care whether the tangent line contains the line , i.e., if the tangent line passes through This is equivalent to
Claim 1. The map is ramified only over points in
By Lemma 1 above we know that the ramification points of are the images of points in By Bezout’s Theorem contains precisely points when counted with multiplicity. By Bezout again, contains precisely the same number of points when considered in Since all intersection points are contained in
Lemma 2. A regular map defined by is ramified precisely over the images of points where Hence it extends to a map ramified over only over those affine ramification points and .
Note that if is ramified over if and only if there exists such that and
Observe that We find that and thus Thus if , then and Conversely if we write for some unit then if and only if
Let be a regular map ramified only over a finite set
Claim 2. If , then there exists such that is ramified only over points in
Let be the maximum degree over of an element in and be the number of elements in of degree We will proceed by double induction on and
Our object is to construct a map such that is ramified only over points in where the number of elements in of degree over is less than and the maximum degree of elements in is less than or equal to
Let be an element of degree over and let be its minimal polynomial. Define by Let be the roots of Note that since the elements are of degree or fewer over
Observe that ramification points of are Moreover, applying a polynomial over cannot increase the degree of an algebraic element over Thus the maximum degree over of elements in is and there are at most such elements since
Claim 3. If then there exists a regular map such that is ramified only over points in
If then clearly there is an automorphism such that
Suppose that We will construct a map such that is ramified over some set of points with
By applying an appropriate automorphism of we may assume that Subsequently we may apply the automorphisms and as necessary to ensure that ; note that each maps
Write Let Let us define by where Observe that which has zeros only at and Thus is ramified only over Continuing inductively and composing we construct a map such that and is ramified only over the values in
Belyi’s Theorem follows from claims 1, 2, and 3.
Example. Consider the elliptic curve defined by the affine equation Note the curve has one point “at infinity,” namely the point Let be the projection away from this point, that is, on the affine patch Note that that this extends to a projection since is nonsingular. This is a slightly different projection than the one in the proof above, but works nicely in this case.
Observe that since Therefore projection is ramified over and the images of the points in where These are the points
The minimal polynomial of over is Thus we consider the map defined by ; note this map is ramified only at We compute We next apply the map defined by which achieves the dual purpose of mapping to and placing Finally, we apply the Belyi map which sends to