**[ Pobierz całość w formacie PDF ]**

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 !
**[ Pobierz całość w formacie PDF ]**