Elkies. .Shimura.Curve.Computations.(2000).[sharethefiles.com]

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primes p such that the point on the (2, 4, 6) curve with coordinate t reduces to
a supersingular point mod p.
Enumeration and arithmetic of covers. When an arithmetic subgroup of
PSL2(R) is commensurable with a triangle group G = Gp,q,r, as was the case
for the � = {2, 3} algebra, any modular cover H/G of H/G (for G �" G a
congruence subgroup) is ramified above only three points on the genus-0 curve
H/G. We readily obtain the ramification data, which leave only finitely many
possibilities for the cover. We noted that, even when there is only one such cover,
actually finding it can be far from straightforward; but much is known about
covers of P1 ramified at three points for instance, the number of such covers
with given Galois group and ramification can be computed by solving equations
in the group (see [Mat]), and the cover is known [Be] to have good reduction
at each prime not dividing the size of the group. But when G, and any group
commensurable with it, has positive genus or more than three elliptic points,
we were forced to introduce additional information about the cover, namely the
existence of an involution exchanging certain preimages of the branch points. In
the examples we gave here (and in several others to be detailed in future work)
this was enough to uniquely determine the cover H/G �! H/G. But there is
as yet no general theory that predicts the number of solutions of this kind of
covering problem. The arithmetic of the solutions is even more mysterious: recall
3
for instance that in our final example the cubic field Q[� ]/(� - 4� + 2) emerged
out of conditions on the cover X0((� ))/X (1) in which that field, and even its
ramified prime 37, are nowhere to be seen.
6 Appendix: Involutions of P1
We collect some facts concerning involutions of the projective line over a field of
characteristic other than 2. We do this from a representation-theoretic point of
view, in the spirit of [FH]. That is, we identify a pair of points ti = (xi : yi) (i =
1, 2) of P1 with a binary quadric, i.e. a one-dimensional space of homogeneous
2
quadratic polynomials Q(X, Y ) = AX2 +2BXY +CY , namely the polynomials
vanishing at the two points; we regard the three-dimensional space V3 of all such
2
polynomials AX2 + 2BXY + CY as a representation of the group SL2 acting
on P1 by unimodular linear transformations of (X, Y ).
An invertible linear transformation of a two-dimensional vector space V2 over
"
any field yields an involution of the projective line P1 = P(V2 ) if and only if
it is not proportional to the identity and its trace vanishes (the first condition
being necessary only in characteristic 2). Over an algebraically closed field of
characteristic other than 2, every involution of P1 has two fixed points, and
any two points are equivalent under the action of PSL2 on P1. It is clear that
the only involution fixing 0, " is t �! -t; it follows that any pair of points
determines a unique involution fixing those two points. Explicitly, if B2 = AC,
2
the involution fixing the distinct roots of AX2 + 2BXY + CY is (X : Y ) �!
44 Noam D. Elkies
(BX +CY : -AX -BY ). Note that the 2-transitivity of PSL2 on P1 also means
that this group acts transitively on the complement in the projective plane PV3
of the conic B2 = AC (and also acts transitively on that conic); indeed it is
well-known that PSL2 is just the special orthogonal group for the discriminant
quadric B2 - AC on V3.
Now let Q1, Q2 " V3 be two polynomials without a common zero. Then there
is a unique involution of P1 switching the roots of Q1 and also of Q2. (If Qi
has a double zero the condition on Qi means that its zero is a fixed point of the
involution.) This can be seen by using the automorphism group Aut(P1) =PGL2
2
to map Qi to XY or Y and noting that the involutions that switch t = 0 with
" are t �! a/t for nonzero a, while the involutions fixing t = " are t �! a - t
for arbitrary a. As before, we regard the involution determined in this way by
Q1, Q2 as an element of PV3. This yields an algebraic map f from (an open set
in) PV3 � PV3, parametrizing Q1, Q2 without common zeros, to PV3. We next
determine this map explicitly.
First we note that this map is covariant under the action of PSL2: we have
f(gQ1, gQ2) = g(f(Q1, Q2)) for any g " PSL2. Next we show that f has degree 1
in each factor. Using the action of PSL2 it is enough to show that if Q1 = XY or
2 2
Y then f is linear as a function of Q2 = AX2 + 2BXY + CY . In the first case,
2
the involution is t �! C/At and its fixed points are the roots of AX2 - CY .
In the second case, the involution is t �! (-2B/A) - t with fixed points t = "
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