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field of degree n > 1 over Q, and any quaternion algebra over that field ramified
at n - 1 of its real places, the group (1) of norm-1 elements of a maximal order
embeds as a discrete subgroup of PSL2(R) = Aut(H), with H/ of finite area
9
Actually this fact is due to Fricke [F1,F2], over a century ago; but Fricke could not
relate G2,3,7 to a quaternion algebra because the arithmetic of quaternion algebras
had yet to be developed.
Shimura Curve Computations 37
given by Shimizu s formula
d3/2K (2)
K
= (-1)n
Area(X (1)) = (N! - 1) K(-1) (N! - 1) (78)
4n-12n 2n-2
!" !"
(from which we obtained (8) by taking K = Q). Thus, in our case of K =
Q(cos 2/7), = { ", " }, the area of H/ (1) is 1/42, so (1) must be iso-
morphic with G2,3,7. From this Shimura deduced [S2, p.83] that for any proper
ideal I " OK his curve X (I) = H/ (I) attains the Hurwitz bound. For in-
stance, if I is the prime ideal !7 above the totally ramified prime 7 of Q then
X (!7) is the Klein curve of genus 3 with automorphism group PSL2(F7) of order
168. The next-smallest example is the ideal !8 above the inert prime 2, which
yields a curve of genus 7 with automorphism group [P]SL2(F8) of order 504.
This curve is also described by Shimura as a known curve , and indeed it first
appears in [F3]; an equivalent curve was studied in detail only a few years before
Shimura by Macbeath [Mac], who does not cite Fricke, and the identification
of Macbeath s curve with Fricke s and with Shimura s X (!8) may first have
been observed by Serre in a 24.vii.1990 letter to Abhyankar. At any rate, we
obtain towers {X (!r)}r>0, {X (!r)}r>0 of unramified abelian extensions which
7 8
are asymptotically optimal over the quadratic extensions of residue fields10 of K
other than F49 and F64 respectively, which are involved in the class field towers
of exponents 7, 2 of the Klein and Macbeath curves over those fields.
These towers are the Galois closures of the covers of X (1) by X0(!r), X0(!r),
7 8
which again may be obtained from the curves X0(!7), X0(!8) together with their
involutions. It turns out that these curves both have genus 0 (indeed the corre-
sponding arithmetic subgroups 0(!7), 0(!8) of (1) are the triangle groups
G3,3,7, G2,7,7 in [T, class X]). The cover X0(!7)/X (1) has the same ramification
data as the degree-8 cover of classical modular curves X0(7)/X(1), and is thus
given by the same rational function
(x4 - 8x3 - 18x2 - 88x7 + 1409)2
7 7 7
t =
21333(9 - x7)
(79)
(x2 - 8x7 - 5)3(x2 + 8x7 + 43)
7 7
= 1 +
21333(9 - x7)
(with the elliptic points of orders 2, 3, 7 at t = 0, 1, ", i.e. t corresponds to
1 - 12-3j). The involution is different, though: it still switches the two simple
"
zeros x7 = -4 -27 of t - 1, but it takes the simple pole x7 = 0 to itself
instead of the septuple pole at x7 = ". Using (89) again we find
19x7 + 711
w! (x7) = . (80)
7
13x7 - 19
10
That is, over the fields of size p2 for primes p = 7 or p a" 1 mod 7, and p6 for other
primes p.
38 Noam D. Elkies
For the degree-9 cover X0(!8)/X (1) we find
(1 - x8)(2x4 + 4x3 + 18x2 + 14x8 + 25)2
8 8 8
t =
27(4x2 + 5x8 + 23)
8
(81)
4(x3 + x2 + 5x8 - 1)3
8 8
= 1 - ,
27(4x2 + 5x8 + 23)
8
with the involution fixing the simple zero x8 = 1 and switching the simple poles,
i.e.
51 - 19x8
w! (x8) = . (82)
8
19 + 13x8
Note that all of these covers and involutions have rational coefficients even
though a priori they are only known to be defined over K. This is possible
because K is a normal extension of Q and the primes !7, !8 used to define
our curves and maps are Galois-invariant. To each of the three real places of K
corresponds a quaternion algebra ramified only at the other two places, and thus
a Shimura curve X (1) with three elliptic points P2, P3, P7 to which we may as-
sign coordinates 0, 1, ". Then Gal(K/Q) permutes these three curves; since we
have chosen rational coordinates for the three distinguished points, any point
on or cover of X (1) defined by a Galois-invariant construction must be fixed
by this action of Galois and so be defined over Q. The same applies to each
of the triangle groups Gp,q,r associated with quaternion algebras over number
fields F properly containing Q, which can be found in cases III through XIX
of Takeuchi s list [T]. In each case, F is Galois over Q, and the finite ramified
places of the quaternion algebra are Galois-invariant. Moreover, even when Gp,q,r
is not (1), it is still related with (1) by a Galois-invariant construction (such
as intersection with 0(!) or adjoining w! or w! for a Galois-invariant prime !
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