3. The Shimura Curves
3.6 The Jacquet-Langlands correspondence
i
2 , i = 1 . . . h.
3.6 The Jacquet-Langlands correspondence
From the Jacquet-Langlands correspondence, we can see a connection between the space of cusp forms on classical modular curves and the space of automorphic forms on Shimura curves X0D(N ).
The definition of Hecke operators on the space of automoprhic forms on Shimura curves X0D(N ) are the same as that of the classical modular forms. We assume that O = O(D, N ) is an Eichler order of level N in an indefinite quaternion algebra of discriminant D. Now fix an imbedding ι : B −→ M (2, R).
Definition 3.6.1. Let p be a prime with p - DN , and α ∈ O be such N (α) = p. Then for an automorphic formf (τ ) of even weight k on Γ = Γ(O), the action of Hecke operator Tponf (τ ) is defined by
Tp: f (τ ) 7→ pk/2−1 X
γ∈Γ\Γι(α)Γ
(det γ)k/2 (cτ + d)kf (γτ ), whereγ = a bc d.
Hecke operators Tn for general n with gcd(n, DN ) = 1 are more complicated.
As in the case of classical modular curves, there exists a basis of Sk(O) consisting of simultaneous eigenforms for all Tn, with (n, DN ) = 1. The Jacquet-Langlands correspondence gives an isomorphism of Hecke modules from Sk(O(D, N )) to the space of cusp forms of weight k and level N which are new at all primes dividing D.
Now let Sk(D, N ) stand for the space of automorphic forms of weight k on Γ(O(D, N )) and simply Sk(M ) = Sk(1, M ), the space of cusp forms of weight k on Γ0(M ). De-note by wm= wm(D, N ) the Atkin-Lehner involution in O(D, N ). Then the Jacquet-Langlands correspondence in our case can be stated as follows.
Proposition 3.6.1 ([12, 18, 33]). We have Sk(D, N ) ' SD−new
k (DN ) :=M
d|N
M
m|Nd
Snewk (dD)[m]
as Hecke modules, where
Snewk (dD)[m] =f (mτ ) : f (τ ) ∈ Snewk (dD) ,
andSnewk (M ) is the subspace of newforms of Sk(M ). Moreover, for a prime p | D, if the action of the Atkin-Lehner involutionwp(1, DN ) on a normalized Hecke eigenform f ∈ SD−new
k (DN ) is wp(1, DN )f = εpf , then the action of wpon the corresponding automorphic form ef ∈ Sk(D, N ) is
wpf = −εe pf .e
According to the Jacquet-Langlands correspondence, we can see that each Hecke eigenform ef for Tpis with the same eigenvalues as the cusp form f .
Chapter 4
Automorphic Forms in Terms of Solutions of Schwarzian
Differential Equations
Let B be an indefinite quaternion algebra of discriminant D over Q. For an Eichler order O of level N , (D, N ) = 1, in B, we let X0D(N ) denote the Shimura curve associated to O. For each divisor m of DN with (m, DN/m) = 1, we let wmdenote the Atkin-Lehner involution on X0D(N ) and WD,N be the group of all Atkin-Lehner involutions. We also let the subgroup of WD,N consisting of wm, m|D, be denoted by WD.
Many properties and theories about classical modular curves can be extended to the case of Shimura curves. In the classical case, many results are relying on the Fourier expansions of modular forms. However, because of the absence of cusps in the case of general Shimura curves (D 6= 1), it is not easy to determine Taylor coefficients of automorphic forms and functions. Therefore, there have been very few results on arithmetic of Shimura curves, and few methods to construct automorphic forms and functions on Shimura curves. One of the few methods uses differential equations sat-isfied by automorphic forms and automorphic functions. (See [2, 6, 33].) The idea is that even though it is difficult to explicitly construct automorphic functions that can be put into practical use, the Schwarzian differential equations associated to automorphic functions in the case of Shimura curves of genus zero can often be determined us-ing analytic information of the automorphic functions and coverus-ings between Shimura curves. Then one can use the solutions of the Schwarzian differential equations in place of automorphic forms to study properties of automorphic forms.
From the result of Yang [33], every automorphic form on a Shimur curve X which is of genus zero can be expressed by the solutions of Schwarzian differential tion associated to X. In view of the significance of Schwarzian differential equa-tions, it is important to determine the Schwarzian differential equation for each of the Shimura curves X0D(N )/G, G < WD,N, of genus zero. In [6], Elkies worked out the Schwarzian equation on X010(1)/W10, X014(1)/W14, and X015(1)/W15. Bayer and
Travesa [2] computed all the Schwarzian differential equations for the Shimura curves X06(1)/G with G < W6. In [33], Yang also gave Schwarzian differential equation on X06(1)/W6and X010(1)/W10from the properties of the automorphic derivatives.
In this chapter, we will consider the cases X0D(N )/WDwhen there exists a square-free integer M > 1 such that X0D(M )/WDhas genus zero. The reason for this restric-tion is that we need addirestric-tional informarestric-tion from coverings between Shimura curves of genus zero in order to completely determine the differential equations. (Note that in [33], a covering between Shimura curves of different levels is also needed in order to compute Hecke operators.) In the process, we also need work out equations for some Shimura curves of genus one and hyperelliptic Shimura curves, which are useful in determining the covering maps between Shimura curves. As a byproduct of our com-putation of coverings X0D(N )/WD→ X0D(1)/WD, we can also determine the values of Hauptmoduls at several CM-points.
In this chapter, we will describe a way to construct automorphic forms on Shimura curves in Section 4.1 . The rest of this chapter is organized as follows. In Section 4.2, we determine all Shimura curves X0D(N )/WDof genus 0, N > 1. In Section 4.3, we will find explicit coverings of X0D(N )/WD → X0D(1)/WD. The equations for Shimura curves and the methods to obtain them given in [8, 9, 14] are important here.
The explicit coverings will be used later. In Section 4.4, we will list the Schwarzian dif-ferential equations for the selected Shimura curves. These results is mainly following the preprint [23].
4.1 Automorphic Forms on Shimura Curves and Schwarzian Differential Equations
Let t(τ ) be a non-constant automorphic function on a Shimura curve X. It is straight-forward to verify that t0(τ ) is a meromorphic automorphic form of weight 2 on X and that the Schwarzian derivative
{t, τ } := t000(τ ) t0(τ ) −3
2
t00(τ ) t0(τ )
2
is a meromorphic automorphic form of weight 4 on X. Thus, the ratio of {t, τ } and t0(τ )2is an automorphic function on X. In particular, if X has genus zero and t(τ ) is a Hauptmodul, i.e., the function t generates the field of automorphic functions on X, then
Q(t) := −{t, τ } 2t0(τ )2
is a rational function of t. In literature [2], given a thrice-differentiable function f of z, the function
D(f, z) := − {f, z}
2f0(z)2 is called the automorphic derivative associated to f .
Now the relation 2Q(t)t0(τ )2+ {t, τ } = 0 can also be written as d2
dt(τ )2t0(τ )1/2+ Q(t)t0(τ )1/2= 0.
In other words, if we consider t0(τ )1/2as a function of t, then t0(τ )1/2is a solution of the differential equation
d2
dt2f + Q(t)f = 0.
Definition 4.1.1. The differential equation d2f /dt2+Q(t)f = 0 is called the Schwarzian differential equation associated to t(τ ).
This differential equation is a Fuchsian differential equation. For each singularity, there is a basis of local solutions of the form
xe(1 + a1x + a2x + · · ·) ,
where e is the local exponent at the singular point. We also remark that this differential equation can be regarded as a normal form for all atomorphic differential equation associated to the group Γ with X = Γ \ h, because it depends only on the chosen of t(τ ).
4.1.1 Automorphic forms on Shimura curves of genus zero
The significance of Schwarzian differential equations can be seen from the following result.
Proposition 4.1.1 ([33, Theorem 4]). Assume that a Shimura curve X has genus zero with elliptic pointsτ1, . . . , τrof orderse1, . . . , er, respectively. Lett(τ ) be a Haupt-modul ofX and set ai = t(τi), i = 1, . . . , r. For a positive even integer k ≥ 4, then a basis forSk(X) is
t0(τ )k/2t(τ )j
r
Y
j=1,aj6=∞
(t(τ ) − aj)−bk(1−1/ej)/2c, j = 0, . . . , dk− 1,
wheredk = dim Sk(X) and it is equal to 1 − k +Pr jbk2
1 − e1
j
c.
Moreover, the automorphic derivative Q(t) satisfies some conditions.
Proposition 4.1.2. Assume that X has genus zero with elliptic points τ1, . . . , τr of ordere1, . . . , er, respectively. Lett(τ ) be a Hauptmodul of X and set ai = t(τi), i = 1, . . . , r. Then the automorphic derivative Q(t) = D(t, τ ) is equal to
Q(t) =1 4
r
X
j=1,aj6=∞
1 − 1/e2j (t − aj)2 +
r
X
j=1,aj6=∞
Bj
t − aj
for some constantsBj. Moreover, ifaj 6= ∞ for all j, then the constants Bjsatisfy
In other words, if we can determine the Schwarzian differential equation associated to a Hauptmodul on a Shimura curve, then we can express automorphic forms of any even weight k on this Shimura curve in terms of solutions of the differential equation.
Corollary 4.1.3. Let X be a Shimura curve of genus zero with elliptic points τ1,. . ., τrof ordere1,. . ., er, respectively. Lett(τ ) be a Hauptmodul of X and set ai= t(τi).
Suppose that{g1, g2} is a basis for the solution space of the Schwarzian differential equation associated tot,
This provides a concrete space that we can use to study properties of automorphic forms. For example, in [33], Yang devised a method to determine Hecke eigenforms in the spaces of automorphic forms, expressed in terms of solutions of Schwarzian differential equations.
Now the upshot is that it is often possible to determine a Schwarzian differen-tial equation without constructing a Hauptmodul first. This is especially true when a Shimura curve of genus zero has three elliptic points. This is due to the well-known fact that a second-order Fuchsian differential equation with precisely three singularities is uniquely determined its local exponents at the three points.
4.1.2 Hypergeometric functions as automorphic forms on Shimura curves
In the case that the Shimura curve of genus 0 has exactly 3 elliptic points, since the number of singularities of the differential equation is 3, the differential equation is es-sentially a hypergeometric differential equation. Then one can express the automorphic forms by using2F1-hypergeometric functions.
To be more precise, when a Shimura curve has signature (0; e1, e2, e3), we let τ1, τ2, τ3be the three elliptic points corresponding to e1, e2, e3. Since X has genus 0,
there exists a unique Hauptmodul t that takes values 0, 1, ∞ at τ1, τ2, τ3, respectively.
According to Proposition 4.1.3, the functions t0(τ )1/2and τ t0(τ )1/2, as functions of t, satisfy the differential equation f00+ Q(t)f = 0, where
Q(t) =1
Combining this with Proposition 4.1.3, we see that every automorphic form on X can be expressed in terms of hypergeometric functions.
Proposition 4.1.1 ([33, Theorem 9]). Assume that a Shimura curve X has signature (0; e1, e2, e3). Let t(τ ) be the Hauptmodul of X with values 0, 1, and ∞ at the elliptic points of ordere1,e2, ande3, respectively. Letk ≥ 4 be an even integer. Then a basis for the space of automorphic forms of weightk on X is given by
t{k(1−1/e1)/2}(1 − t){k(1−1/e2)/2}tj
2F1(a, b; c; t) + Ct1/e12F1(a0, b0, c0; t)k , j = 0, . . . , bk(1 − 1/e1)/2c + bk(1 − 1/e2)/2c + bk(1 − 1/e3)/2c − k, for some constantC, where for a rational number x, we let {x} denote the fractional part of x,
a = 1
In [24], Yang and the author of the present paper obtained several new algebraic transformation of2F1-hypergeometric functions by interpreting identities among hy-pergeometric functions as identities among automorphic forms on different Shimura curves. In chapter 6, we will introduce how we obtain algebraic transformations of
2F1-Hypergeometric functions.
4.1.3 Transformation laws of automorphic derivatives
For general Shimura curves, the following properties of Schwarzian differential tions and automorphic derivatives are very useful in determining the differential equa-tions.
Proposition 4.1.4. [33] Automorphic derivatives have the following properties.
1. D((az + b)/(cz + d), z) = 0 for all a bc d ∈ GL(2, C).
2. D(g ◦ f, z) = D(g, f (z)) + D(f, z)/(dg/df )2.
Proposition 4.1.5. [33] Let t(τ ) be a Hauptmodul for a Shimura curve X of genus 0. Let R(x) ∈ C(x) be the rational function such that the automorphic derivative Q(t) = D(t, τ ) is equal to R(t). Assume that γ is an element of GL(2, R) normalizing the orderO associated to X and let σ be the automorphism of X induced by γ. If σ : t 7→ (at + b)/(ct + d), then R(x) satisfies
Proof. We shall compute D(t(γτ ), τ ) in two ways. By Proposition 4.1.4, we have D(t(γτ ), τ ) = D at(τ ) + b On the other hand, by the same proposition, we also have
D(t(γτ ), τ ) = D(t(γτ ), γτ ) + D(γτ, τ )
(dt(γτ )/dγτ )2 = R(t(γτ )) = R at + b ct + d
. Comparing the two expressions, we get the formula.
4.2 Shimura Curves of Genus Zero
From now on, let us consider the Shimura curves X0D(N ) and fix the notation WD = WD,1. In this section, we will determine all pairs of integers (D, N ), D, N > 1, such that X0D(N )/WDhas genus 0, where N is a squarefree integer. We will need explicit coverings X0D(N )/WD→ X0D(1)/WDin order to determine Schwarzian differential equations.
A formula for the genus of X0D(N )/G, G < WD,N, will involve the numbers of CM points of certain discriminants. For the goal of this section, we only need to know the number of CM-points associated to K = Q(√
−m) with m|D of discriminant −3, dK, or 4dKin the case dK≡ 1 mod 4.
Lemma 4.2.1 ([15], or Section 3.3 and Section 3.2.2). For m|D or m = 3, let dK
denote the discriminant of the fieldK = Q(√
−m). We have
Also, form|D with m ≡ 3 mod 4, we have
Hereh(d) is the class number of the imaginary quadratic order of discriminant d.
Lemma 4.2.2. The complete list of integers (D, N ) with D, N > 1 such that the Shimura curveX0D(N )/WDhas genus zero, is
(6, 5), (6, 7), (6, 13), (10, 3), (10, 7), (14, 3), (14, 5), (15, 2), (15, 4), (21, 2), (26, 3), (35, 2), (39, 2).
Proof. Let Γ be a congruence Fuchsian subgroup of SL(2, R). (See [13] for the def-inition of a congruence Fuchsian subgroup. The groups considered here are all con-gruence Fuchsian subgroups.) A famous result of Selberg [16] stated that if Γ is a congruence subgroup of SL(2, Z), then the first eigenvalue λ1of the Laplace operator on the space of square-integrable function on Γ\h is not less than 3/16. By combining this result with the Jacquet-Langlands correspondence, Vign´eras [27] showed that the same inequality also holds for congruence Fuchsian subgroups coming from indefinite quaternion algebras over Q of discriminant not equal to 1.
On the other hand, Zograf [34] showed that if the area Vol(Γ\h) is at least 16(g(Γ)+
1), then λ1 < 4(g(Γ) + 1)/Vol(Γ\h). Here g(Γ) denotes the genus of Γ and the area is normalized such that A(SL(2, Z)\h) = 1/6. Combining Selberg’s inequality and Zograf’s result, one sees that if a congruence Fuchsian subgroup has genus 0, then the area must be less than 64/3.
Now recall that the area of X0D(N ) is given by
This immediately shows that if the number of prime factors of D is at least 6, then the genus of X0D(N )/WD cannot be 0 for any N ≥ 2. Also, if D = pq is a product of two primes such that (p − 1)(q − 1) > 512/3, then X0D(N )/WDmust have a positive
genus for any N ≥ 2. A similar bounds exists for the case D has 4 prime factors. This leaves finitely many cases to check.
Note that the genus of a Shimura X is given by
g(X) = 1 +Vol(X)
2 −1
2
r
X
i=1
1 − 1
ei
,
where the sum runs through all elliptic points with eibeing their respective orders. For X = X0D(N )/WD, by Lemma 3.4.1, we have
g(X) = 1 + Vol(X)
2 −1
4 X
m|D,m6=1,3
1
2r−1(#CM(−4m) + #CM(−m))
−
1
4 · 2r#CM(−4), if 2 - D, 3
8 · 2r−1#CM(−4), if 2|D
−
1
3 · 2r#CM(−3), if 3 - D,
1
4 · 2r−1#CM(−12) + 5
12 · 2r−1#CM(−3)
, if 3|D, where r is the number of prime divisors of D. (Of course, if d is not a discriminant, then we simply let CM(d) be the empty set.)
Using the Selberg-Zograf bound, the genus formula in the paragraph above and Lemma 4.2.1, we check case by case that the pairs of integers given in the lemma are the only cases where X0D(N )/WD, N > 1, has genus zero.
We now tabulate all Shimura curves X0D(M )/WDof genus 0 for integers D that appear in the lemma. We will also give a description of their elliptic points. These Shimura curves are the curves that we wish to determine their Schwarzian differential equations. Here vjdenotes the number of elliptic points of order j on X0D(M )/WD. Here we also let CM(−m) denote the set of points on X0D(N )/WDthat are the image of CM points of discriminants −m under the covering X0D(N ) → X0D(N )/WD. The number n in CM(−m)×nmeans the number of elements in CM(−m) is n. If n = 1, we omit this annotation.
D, N v2, v3, v4, v6 elliptic points
6, 1 1, 0, 1, 1 CM(−3), CM(−4), CM(−24) 6, 5 2, 0, 2, 0 CM(−4)×2, CM(−24)×2 6, 7 2, 0, 0, 2 CM(−3)×2, CM(−24)×2 6, 13 0, 0, 2, 2 CM(−3)×2, CM(−4)×2
10, 1 3, 1, 0, 0 CM(−3), CM(−8), CM(−20), CM(−40) 10, 3 4, 1, 0, 0 CM(−3), CM(−8)×2, CM(−20)×2 10, 7 4, 2, 0, 0 CM(−3)×2, CM(−20)×2, CM(−40)×2 14, 1 3, 0, 1, 0 CM(−4), CM(−8), CM(−56)×2 14, 3 6, 0, 0, 0 CM(−8)×2, CM(−56)×4 14, 5 4, 0, 2, 0 CM(−4)×2, CM(−56)×4
15, 1 3, 0, 0, 1 CM(−3), CM(−12), CM(−15), CM(−60) 15, 2 6, 0, 0, 0 CM(−12)×2, CM(−15)×2, CM(−60)×2 15, 4 8, 0, 0, 0 CM(−12)×2, CM(−15)×2, CM(−60)×4 21, 1 5, 0, 0, 0 CM(−4), CM(−7), CM(−28), CM(−84)×2 21, 2 7, 0, 0, 0 CM(−4), CM(−7)×2, CM(−28)×2, CM(−84)×2 26, 1 5, 0, 0, 0 CM(−8), CM(−52), CM(−104)×3
26, 3 8, 0, 0, 0 CM(−8)×2, CM(−104)×6
35, 1 6, 0, 0, 0 CM(−7), CM(−28), CM(−35), CM(−140)×3 35, 2 10, 0, 0, 0 CM(−7)×2, CM(−28)×2, CM(−140)×6 39, 1 6, 0, 0, 0 CM(−52)×2, CM(−39)×2, CM(−156)×2 39, 2 10, 0, 0, 0 CM(−52)×2, CM(−39)×4, CM(−156)×4
4.3 Coverings of Shimura Curves
The goal of this section is to obtain explicit coverings of X0D(N )/WD→ X0D(1)/WD for pairs of D and N given in Lemma 4.2.2. That is, we wish to find a Hauptmodul t1of X0D(1)/WD, a Hauptmodul tN of X0D(N )/WD, and the relation between them.
Of course, there are infinitely many choice for t1and tN. For X0D(N )/WD, we will choose tN such that the Atkin-Lehner involution wN acts by wN : tN 7→ −tN. This will make the determination of Schwarzian differential equation simpler.
Case D = 6 In the case D = 6, all the coverings X06(N )/W6 → X06(1)/W6, N = 5, 7, 13, are already given in [6]. Here we just modify the tN in [6] such that the new tN satisfies wN : tN 7→ −tN.
Lemma 4.3.1 ([6]). 1. There is a Hauptmodult1forX06(1)/W6that takes values 0, 1, and ∞ at the CM-points of discriminants −24, −4, and −3, respectively.
2. There is a Hauptmodult = t5forX06(5)/W6that takes values±i/8 and ±√
−6/3 at the CM-points of discriminants−4 and −24, respectively. The relation be-tweent1andt is
t1= (2 + 3t2)(34 − 117t + 1824t2)2
125(1 + 6t)6 = 1 +27(1 + 64t2)(3 − 7t)4 125(1 + 6t)6 .
The Atkin-Lehner involutionw5acts byw5: t 7→ −t.
3. There is a Hauptmodult = t7 forX06(7)/W6 that takes values±√
−3/9 and
±√
−6/8 at the CM-points of discriminants −3 and −24, respectively. The relation betweent1andt is
t1= −(3 + 32t2)(78 − 396t + 1963t2− 12312t3)2 4(1 + 27t2)(3 + 10t)6
The Atkin-Lehner involutionw7acts byw7: t 7→ −t.
4. There is a Hauptmodult = t13forX06(13)/W6that takes values±4√
−3/9 and
±3i/4 at the CM-points of discriminants −3 and −4, respectively. The relation betweent1andt is
t1= 1 − 27(9 + 16t2)(144 − 98t + 246t2− 161t3)4 16(16 + 27t2)(30 + 3t + 55t2)6 . The Atkin-Lehner involutionw13acts byw13: t 7→ −t.
Proof. In [6], Elkies already showed that explicit coverings of X06(N )/W6→ X06(1)/W6, N = 5, 7, 13, are given by
t1= 1 + 135s4+ 324s5+ 540s6, w5: s 7−→ 42 − 55s 55 + 300s, t1= −(4s2+ 4s + 25)(2s3− 3s2+ 12s − 2)2
108(7s2− 8s + 37) , w7: s 7−→116 − 9s 9 + 20s, and
t1=(s7− 50s6+ 63s5− 5040s4+ 783s3− 168426s2− 6831s − 1864404)2 4(7s2+ 2s + 247)(s2+ 39)6
with
w13: s 7−→5s + 72 2s − 5 , respectively. Choosing t such that
s = 7t − 3
30t + 5, s = −29t + 6
10t + 3 , s = −8t + 9 2t + 1 , respectively, we get the lemma.
Case D = 10 The covering X010(3)/W10 → X010(1)/W10 has also been given in [6]. Here we mainly work on the case N = 7.
Lemma 4.3.2. 1. There is a Hauptmodult1forX010(1)/W10that takes values0,
∞, 2, and 27 at the CM-points of discriminants −3, −8, −20, and −40, respec-tively.
2. There is a Hauptmodult = t3forX010(3)/W10that takes values0, ±1/4√
−2,
±1/√
−5 at the CM-points of discriminants −3, −8, and −20, respectively. The relation betweent1andt is
t1= 108t(1 − 2t)3
(1 + 32t2)(1 + 7t)2 = 2 −2(1 + 5t2)(1 − 20t)2 (1 + 32t2)(1 + 7t)2 . The Atkin-Lehner involutionw3acts byw3: t 7→ −t.
3. There is a Hauptmodul t = t7for X010(7)/W10 that takes values±1/3√
−3,
±1/2√
−5, and ±√
−10/16 at the CM-points of discriminants −3, −20, and
−40, respectively. The relation between t1andt is t1= 8(1 + 27t2)(2 − 3t + 44t2)3
7(1 + 4t + 55t2+ 102t3+ 736t4)2. The Atkin-Lehner involutionw7acts byw7: t 7→ −t.
Proof. In [6], it is shown that an explicit covering X010(3)/W10 → X010(1)/W10is given by
t1= 216(s − 1)3 (s + 1)2(9s2− 10s + 17)
with w3: s 7→ 10/9 − s. Let t be the Hauptmodul of X010(1)/W10with s = 2
9t+5 9.
Then the relation of t1and t and the action of w3are given as in the lemma.
We next consider the case N = 7. According to Theorem 3.4 of [9], an equation for X010(7) is given by
y2= −27x4− 40x3+ 6x2+ 40x − 27. (4.1) The actions of the Atkin-Lehner involutions on this model of X010(7) are given by
w70: (x, y) 7−→ (x, −y), w5: (x, y) 7−→
−1 x, −y
x2
, and
w10: (x, y) 7−→ 2x + 1 x − 2 , 5y
(x − 2)2
.
Since CM(−20) are fixed points of w5, their coordinates on (4.1) are (i, ±2√ 5(1 + 2i)) and (−i, ±2√
5(1 − 2i)). Likewise, we find that CM(−40) have coordinates
(2 +√ the method of [9], we know that the two points at infinity are CM-points of discrimi-nant −3. Thus, the coordinates of CM(−3) are ∞, (0, ±3√
On this equation for X0(10)(7)/hw10i, the actions of the Atkin-Lehner involutions are given by The coordinates of CM(−3) are the two points at ∞ and (±3√
−3/2, −1/2). Also, the coordinates of CM(−20) are (±2√
−5, 0), and the coordinates of CM(−40) are (±8√ of CM-points of discriminants −3, −20, and −40 are ±1/3√
−3, ±1/2√
−5, and
±√
−10/16, respectively. It follows that the relation between t1and t is t1= A(1 + 27t2)(1 + a1t + a2t2)3 (or the same expression with t replaced by −t). This proves the lemma.
Case D = 14 The case D = 14 is also worked out in [6]. Here we only need to make a change of variable so that wN acts by wN : tN → −tN.
Lemma 4.3.3 ([6]). 1. There is a Hauptmodult1forX014(1)/W14that takes values
∞, 0, and (−13 ± 7√
−7)/32 at CM-points of discriminants −4, −8, and −56, respectively.
2. There is a Hauptmodult = t3forX014(3)/W14that takes values±1/√
−2 and (±9√
−7 ± 4√
−14)/49 at CM-points of discriminants −8 and −56, respec-tively. The relation betweent1andt is
t1=4(1 + 2t2)(1 − 5t)2 9(1 + t)4 . The Atkin-Lehner involutionw3acts byw3: t 7→ −t.
3. There is a Hauptmodul t = t5 for X014(5)/W14 that takes values±i/4 and (±5√
−7 ± 4√
−14)/7 at CM-points of discriminants −4 and −56, respectively.
The relation betweent1andt is
t1= −5(1 − t + 17t2− 13t3)2 (1 + 16t2)(1 + 3t)4 . The Atkin-Lehner involutionw5acts byw5: t 7→ −t.
Proof. In [6], it is shown that explicit coverings X014(N )/W14 → X014(1)/W14can be given by
t1= 1
27(s4+ 2s3+ 9s2), w3: s 7−→ 5 − 2s 2 + s and
t1= −(256s3+ 224s2+ 232s + 217)2
50000(s2+ 1) , w5: s 7−→ 24 − 7s 7 + 24s, respectively. Choosing t with
s = 1 − 5t
1 + t, s = 3 − 16t 4 + 12t, respectively, we get the lemma.
Case D = 15 An explicit covering X015(2)/W15 → X015(1)/W15 is given in [6].
Here we only need to make a change of variable so that wN acts by wN : tN → −tN. Lemma 4.3.4. 1. There is a Hauptmodul forX015(1)/W15that takes values∞, 0, 81, and 1 at CM-points of discriminants −3, −12, −15, and −60, respectively.
2. There is a Hauptmodul t2 for X015(2)/W15 that takes values±1, ±√
−15/3, and±1/5 at CM-points of discriminant −12, −15, and −60, respectively. The relation betweent1andt2is
t1= 27(1 − t2)(1 − 3t2)2
2(1 + t2)3 = 1+(1 − 5t2)(5 − 7t2)2
2(1 + t2)3 = 81−27(1 + 5t2)(5 + 3t22) 2(1 + t2)3 . The Atkin-Lehner involutionw2acts byw2: t27→ −t2.
Proof. In [6], an explicit covering X015(2)/W15→ X015(1)/W15is given by t1=1
4s(s − 3)2, w2: s 7−→36 s . Choosing a Hauptmodul t for X015(2)/W15with
s = 6 − 6t 1 + t , we establish the claim about X015(2)/W15.
Case D = 21 We will need an equation for some Atkin-Lehner quotient of X021(2) in order to determine the coordinates of elliptic points on X021(2).
Lemma 4.3.5. An equation for X021(2)/hw21i is y2 = (x + 12)(x2 − 7x + 28).
Moreover, the action of the Atkin-Lehner involutionw3= w7on this curve is given by (x, y) 7→ (x, −y). Also, the two rational points ∞ and (−12, 0) are the CM-points of discriminants−28, and the other two 2-torsion points ((7 ± 3√
−7)/2, 0) are the CM-points of discriminant−7.
Proof. We follow the methods of [9]. The Shimura curve X021(2)/hw21i has genus 1. By [9, Lemma 5.10], the two CM-points of discriminant −28 are Q-rational points on this curve. Thus, X021(2)/hw21i is an elliptic curve over Q. Now in the space S2(Γ0(42))21-newthe unique Hecke eigenform with +-eigenvalue for w21is coming from the newform space of S2(Γ0(42)). Therefore, the elliptic curve X021(2)/hw21i has conductor 42. Using the Cerednik-Drinfeld theory of p-adic uniformization of Shimura curves, we find that the types of singular fibers at primes of bad reduction of X021(2)/hw21i agree with those of the elliptic curve 42A1, in Cremona’s notation. The global minimal model of the elliptic curve 42A1 is y2+ xy + y = x3+ x2− 4x + 5.
With a simple change of variables, we write it as y2= (x + 12)(x2− 7x + 28).
Now the covering X021(2)/hw21i → X021(2)/W21 is ramified at the two CM-points of discriminant −7 and the two CM-CM-points of discriminant −28. If we let one of the CM-points of discriminant −28 be the point at infinity, then an equation for X021(2)/hw21i is of the form y2= f (x) for some polynomial f (x) = x3+ · · · of de-gree 3 in Q[x] with the Atkin-Lehner involution w3acting by (x, y) 7→ (x, −y). Up to a transformation of the form x 7→ ax + b, this polynomial f (x) must be the polynomial (x + 12)(x2− 7x + 28). This proves the lemma.
Remark 4.3.6. According to Cremona’s table of elliptic curves [3], the elliptic curve 42A1 has8 rational points. Thus, X021(2)/hw21i also has 8 Q-rational points. Two of them are the CM-points of discriminant−28 mentioned above. The rest of Q-rational points consist of two CM-points of discriminant−4 and four CM-points of discriminant
−16.
Lemma 4.3.7. There is a Hauptmodul t1forX021(1)/W21that takes values49, 0, ∞, and(47 ± 8√
−3)/7 at CM-points of discriminants −4, −7, −28, and −84, respec-tively.
Also, there is a Hauptmodult = t2forX021(2)/W21that takes values0, ±1/3√
−7,
±1, and ±1/3√
−3 at CM-points of discriminants −4, −7, −28, and −84, respec-tively. The relation betweent1andt is
t1=49(1 + t)(1 + 63t2)
(1 − t)(1 − 15t)2 = 49 + 1568t(1 − 3t)2 (1 − t)(1 − 15t)2. The Atkin-Lehner involutionw2acts byw2: t 7→ −t.
Proof. According to [9], an equation for X021(1) is given by y2= −7x4+ 94x2− 343 with the actions of the Atkin-Lehner involutions given by
w3: (x, y) 7−→ (−x, −y), w7: (x, y) 7−→ (−x, y), w21: (x, y) 7−→ (x, −y).
The Atkin-Lehner involution w7fixes the two points at ∞ and (0, ±7√
−7). Since the equation has a symmetry (x, y) 7−→ (7/x, 7y/x2), we might as well assume that the two points (0, ±7√
−7) are the CM-points of discriminant −7 and the two points at infinity are the CM-points of discriminant −28. Moreover, the four points with y = 0 correspond to the four CM-points of discriminant −84.
Since w3acts by (x, y) → (−x, −y), an equation for X021(1)/hw3i is y2= −7x3+ 94x2 − 343x, where the covering X021(1) → X021(1)/hw3i is given by (x, y) 7→
(x2, xy). Then t1 = x generates the function field of X021/W21. The values of t1
at the CM-points of discriminants −7, −28, and −84 are 0, ∞, and (47 ± 8√
at the CM-points of discriminants −7, −28, and −84 are 0, ∞, and (47 ± 8√