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final results of our computations of modular curves and CM points.
5.1 The case £ = {2, 7}
We generate A by elements b, g with
b2 + 2 = g2 - 7 = bg + gb = 0, (64)
7
On the other hand the factor 23 in (57) was a matter of convenience, to make the
four elliptic points integral.
Shimura Curve Computations 31
Table 3
|D| D0 D1 A B proved?
3 1 1 0 1 Y
8 8 1 1 0 Y
20 20 1 2 1 Y
40 40 1 27 1 Y
52 4 13 -54 25 N
120 40 3 -27 49 Y
35 5 7 64 7 Y
27 1 33 -192 25 Y
72 8 32 125 147 Y
43 1 43 1728 1225 N
180 20 32 -2662 169 Y
88 8 11 3375 98 N
115 5 23 13824 3887 N
280 40 7 35937 7406 N
67 1 67 -216000 8281 N
148 4 37 71874 207025 N
340 20 17 657018 41209 N
520 40 13 658503 11257064 N
232 8 29 176558481 2592100 N
760 40 19 13772224773 237375649 N
163 1 163 -2299968000 6692712481 N
and a maximal order O by Z[b, g] together with (1+b+g)/2 (and b(1+g)/2). By
" "
(9), the curve X (1) has hyperbolic area 1/4. Since “ (1) is not a triangle group
"
(again by [T]), we again conclude by (10) that X (1) has exactly four elliptic
"
points, this time of orders 2, 2, 2, 4. We find in “ (1) the elements of finite order
s2 = [b], s 2 = [7b - 2g - bg], s 2 = [7b + 2g - bg], s4 = [1 + 2b + g] (65)
[NB 7b ± 2g - bg " 2O] of orders 2, 2, 2, 4 with s2s 2s 2s4 = 1, and conclude that
"
s2, s 2, s 2, s4 generate “ (1) with relations determined by s2 = s 22 = s 2 2 = s4 =
2 4
"
s2s 2s 2s4 = 1. None of these is in “ (1): the representatives b, 1 + 2b + g of s2, s4
have norm 2, while s 2, s 2 have representatives (7b ± 2g - bg)/2 of norm 14. The
discriminants of s4, s2, s 2, s 2 are -4, -8, -56, -56; note that -56 is not among
the idoneal discriminants (discriminants of imaginary quadratic fields with
class group (Z/2)r), and thus that the elliptic fixed points P2, P2 of s 2, s 2 are
"
quadratic conjugates on X (1).
"
Again we use the involution w3 on the modular curve X0 (3) to simultaneously
"
determine the relative position of the elliptic points P4, P2, P2, P2 on X (1) and
" "
the modular cover X0 (3) ’! X (1), and then to obtain a Schwarzian equation
" "
on X (1). Clearly P4 is completely ramified in X0 (3). Since -8 and -56 are
"
quadratic residues of 3, each of P2, P2, P2 has ramification type 211. Thus X0 (3)
is a rational curve with six elliptic points all of index 2, and we may choose
" "
coordinates t, x on X (1), X0 (3) such that t(P4) = ", t(P2) = 0, and x = ",
x = 0 at the quadruple pole and double zero respectively of t.
32 Noam D. Elkies
Table 4
|D| D0 D1 |A| B |A - 2B| |A - 27B|
3 1 1 0 1 2 33
8 8 1 1 0 1 1
20 20 1 2 1 0 52
40 40 1 33 1 52 0
52 4 13 2·33 52 2313 36
120 40 3 33 72 53 2·3352
35 5 7 26 7 2·52 53
27 1 33 263 52 2·112 1723
72 8 32 53 723 132 22312
43 1 43 2633 5272 2·192 3643
180 20 32 2·113 132 23533 52172
88 8 11 3353 2·72 17211 36
115 5 23 2933 13223 2·52112 3653
280 40 7 33113 2·2327 53132 3852
67 1 67 263353 72132 2·112312 3867
148 4 37 2·33113 5272132 2517237 38292
340 20 17 2·33233 72292 23521327 3654
520 40 13 33293 237247213 54112172 385243
232 8 29 33113173 225272232 132192532 3671229
760 40 19 33173473 72312712 5211213237219 2·3853672
163 1 163 293353113 72132292312 2·192592792 36172732163
"
We next determine the action of w3 on the elliptic points of X0 (3). Necessarily
the simple preimages of P2 parametrize two 3-isogenies from P2 to itself. On
the other hand the simple preimages of P2 parametrize two 3-isogenies from
that to P2 and vice versa, because the squares of the primes above 3 in
"point
Q( -14) are not principal. Therefore w3 exchanges the simple preimages of P2
but takes each of the two simple points above P2 to one above P2 and vice versa.
So again we have a one-parameter family of degree-4 functions on P1, and
a single condition in the existence of the involution w3; but this time it turns
out that there are (up to scaling the coordinates t, x) two ways to satisfy this
condition:
1 1 - x
t = (x4 + 4x3 + 6x2), w3(x) = , P2, P2 : t2 - 3t + 3 = 0 (66)
3 1 + x
and
1 5 - 2x
t = (x4 + 2x3 + 9x2), w3(x) = , P2, P2 : 16t2 + 13t + 8 = 0. (67)
27 2 + x
"
How to choose the correct one? We could consider the next modular curve X0 (5)
and its involution to obtain a new condition that would be satisfied by only one of
(66,67). Fortunately we can circumvent this laborious calculation by noting that
the Fuchsian group associated with (66) is commensurable with a triangle group,
since its three elliptic points of index 2 are the roots of (1 - t)3 = 1 and are thus
Shimura Curve Computations 33
permuted by a 3-cycle that fixes the fourth elliptic point t = ". The quotient
by that 3-cycle is a curve parametrized by (1 - t)3 with elliptic points of order
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