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Shimura Curve Computations 27
Table 2
|D| D0 D1 |A| B |C| r
84 12 7 132 33 2272 1.19410
40 8 5 23172 53 37 0.80487
51 3 17 3417 210 74 0.84419
19 1 19 13219 210 37 0.90424
120 24 5 24192 3353 74 0.95729
52 4 13 23213 56 2237 1.00276
132 12 11 34132 56 24112 0.87817
75 3 52 172232 210335 114 0.98579
168 24 7 2335192 56 11472 0.79278
88 8 11 25172412 56113 3774 0.86307
43 1 43 19237243 21056 3774 0.92839
228 12 19 132172372 3656 2674192 0.96018
123 3 41 3413223241 21056 74194 0.90513
100 4 52 192232472 116 2437745 0.88998
147 3 72 172412472 21033567 114234 0.96132
312 24 13 243517243213 56116 74234 0.83432
67 1 67 13243261267 21656 3774114 0.89267
148 4 37 13247271237 56176 223774114 0.94008
372 12 31 132232372612 3356116 2274194312 0.99029
408 24 17 26132192432672 3656173 74114314 0.88352
267 3 89 3613217219271289 21656116 74314434 0.87610
232 8 29 231321724128921132 56236293 3774114194 0.91700
708 12 59 341321922323724121092 56176296 2874114474592 0.91518
163 1 163 132672109213921572163 21056116176 31174194234 0.90013
"
genus and number of elliptic points of X (1), X (1), but not the generators of
"
“ (1), “ (1), are already tabulated in [V, Ch.IV:2].)
" "
4.2 Shimura modular curves X0 (l), in particular X0 (3)
The elliptic elements s3, s2, s 2, s 2 have discriminants -3, -8, -20, -40. Thus the
"
curve X0 (l) has genus
1 -3 -2 -5 -10
"
g(X0 (l)) = l - 4 - 3 - 3 - 3 . (54)
12 l l l l
Again we tabulate this for l
l 3 7 11 13 17 19 23 29 31 37 41 43 47
"
g(X0 (l)) 0 0 1 1 2 1 2 3 3 3 3 3 4
"
Since g(X0 (l)) e" (l - 13)/12, the cases l = 3, 7 of genus 0 occurring in this
table are the only ones. We next find an explicit rational functions of degree 4
" "
on P1 that realizes the cover X0 (3)/X0 (1), and determine the involution w3.
" "
The curve X0 (3) is a degree-4 cover of X (1) with Galois group PGL2(F3)
and cycle structures 31, 211, 211, 22 over the elliptic points P3 , P2 , P2, P2 . Thus
28 Noam D. Elkies
" "
there are coordinates Ä, x on X (1), X0 (3) such that Ä (x) = (x2 - c)2/(x - 1)3
for some c. To determine the parameter c, we use the fact that w3 fixes the
simple pole x = " and takes each simple preimage of the 211 points P2 , P2 to
the other simple preimage of the same point. That is,
dx
(x2 - c)-1(x - 1)4 = x2 - 4x + 3c (55)
dt
must have distinct roots xi (i = 1, 2) that yield quadratic polynomials
(x - 1)3(Ä (x) - Ä (xi))
(56)
(x - xi)2
with the same x coefficient. We find that this happens only for c = -5/3, i.e.
that Ä = (3x2 + 5)2/9(x - 1)3. For future use it will prove convenient to use
63 (6x - 6)3
t = = , (57)
9Ä + 8 (x + 1)2(9x2 - 10x + 17)
10
with w3(x) = -x. [Smaller coefficients can be obtained by letting x = 1+2/x ,
9
Ä = 2t /9, when t = (2x 2 + 3x + 3)2/x and w3(x ) = -9x /(4x + 9). But our
choice of x will simplify the computation of the Schwarzian equation, while the
choice of t will turn out to be the correct one 3-adically.] The elliptic points are
then P3 : t = 0, P2 : t = 27, and P2, P2 : t = ", 2. In fact the information so
far does not exclude the possibility that the pole of t might be at P2 instead
of P2; that in fact t(P2) = ", t(P2) = 2 and not the other way around can be
"
seen from the order of the elliptic points on the real locus of X (1), or (once we
compute the Schwarzian equation) checked using the supersingular test.
" "
4.3 CM points on X (1) via X0 (3) and w3
From w3 we obtain five further CM points. Three of these are 3-isogenous to
known elliptic points: w3 takes the triple zero x = 1 of t to x = 1/9, which gives
us t = -192/25, the point 3-isogenous to P3 with discriminant -27; likewise w3
takes the double root x = 5 and double pole x = -1 of t - 2 to x = -35/9, 19/9
and thus to t = -2662/169 and t = 125/147, the points 3-isogenous to t = 2 and
t = " and thus (once these points are identified with P2 and P2) of discriminants
-180 and -72. One new CM point comes from the other fixed point x = 5/9
of w3, which yields t = -27/49 of discriminant -120. Finally the remaining
solutions of t(x) = t(w3(x)) are the roots of 9x2 - 10x + 65; the resulting CM
point t = 64/7, with two 3-isogenies to itself, turns out to have discriminant -35.
"
4.4 The Schwarzian equation on X (1)
"
We can take the Schwarzian equation on X (1) to be of the form
t(t - 2)(t - 27)f + (At2 + Bt + C)f + (Dt + E) = 0. (58)
Shimura Curve Computations 29
The coefficients A, B, C, D are then forced by the indices of the elliptic points.
Near t = 0, the solutions of (58) must be generated by functions with leading
terms 1 and t1/3; near t = 2 (t = 27), by functions with leading terms 1 and
(t - 2)1/2 (resp. (t - 27)1/2); and at infinity, by functions with leading terms [ Pobierz całość w formacie PDF ]

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