KI-ICHIRO HASHIMOTO AND FANG-TING TU
ABSTRACT. In this article, we are studying the arithmetic properties of elliptic Q-curves related to Modularity Theorem. We will give a generic family of (−1)-minimal Q-curves of degree 2, families of (±1)-minimal Q-curves of degree 2, and family of genus 2 curves defined over Q whose Jacobians are of GL2-type with
√
−2 multiplication. Also, we will give some explicit equations of curves whose Jacobians are associate to cer-tain Shimura abelian surfaces.
1. INTRODUCTION
An elliptic curve E defined over a number field is called Q-curve if it is isogenous over ¯Q to any of its Galois conjugatesσE, σ ∈ Gal( ¯
Q/Q). If E is a Q-curves over a quadratic field K, then by restriction of scalars we obtain an abelian surface ResK/Q(E) over Q. In this paper we shall study some basic problems from the computational aspect on Q-curves over quadratic fields. More precisely, the problems with which we shall be concerned are stated as
(i) When is ResK/Q(E) of GL2-type ?
(ii) For a given quadratic field K, find (all) E/K satisfying (i).
(iii) For a Q-curve E/K of (ii), find a genus two curve C over Q such that Jac(C) is isogenous to ResK/Q(E).
Here we recall that an abelian variety A over Q is called of GL2-type if the
Q-algebra generated by the endomorphisms defined over Q contains a field of degree equal to dim(A).
An answer to (i) is now well known : ResK/Q(E) is of GL2-type if and
only if it is minimal. Here a Q-curve E over a quadratic field K is called minimal, if the non-trivial isogeny φσ : E → σE (σ 6= id) is also defined
Date: September 10, 2012.
This work is partial supported by National Center for Theoretical Sciences, and Shing-Tung Yau Center.
over K. Note that we have then
σµ
σ◦µσ = [εd] with ε = ±1,
where d is the degree of φσ which is chosen to be minimum among all
possible values. Hence the set of minimal Q-curves are divided into two classes by the sign of ε; accordingly a Q-curve is called (+1) minimal if ε = +1 and (−1) minimal if ε = −1.
The problems (ii), (iii) for (+1) minimal Q-curves were studied by sev-eral authors (c.f. [1, 4] ), and answers to them were given for some qua-dratic field K. For (−1)-minimal Q-curves, on the other hand, no system-atic study seems to have been done yet. We do not know, for example, over which quadratic field K does there exist a (−1)-minimal Q-curve over K.
Another problem closely related to (iii) is the modularity of minimal Q-curves. Recall that an elliptic curve E over Q i s called modular, if it is covered by the modular curve X1(N ). The modularity of Q-curves was
conjectured by Ribet [11, 12]) and proved in most cases by [3, 5, 8]. The general case, as well as the modularity of abelian varieties of GL2-type, has
been proved by recent works of Khare (c.f. [9]) as a special case Serre’s conjecture. One can then ask the following problem.
(iii)’ For each normalized Hecke eigenform f ∈ S2(Γ1(N )) whose Fourier
coefficients generate a quadratic field, find a curve C of genus two over Q such that Jac(C) is Q-isogenous to Af, Shimura’s abelian
surface.
We shall discuss these problems for Q-curves of degree d = 2. This means that the field generated by the Fourier coefficients of f corresponding to E is Q(√±2). One of our aims is to give answers to the problem (iii) and (iii)’ for each of the cusp form f ∈ S2(Γ1(N )) satisfying this condition
with small levels N (N < 1000).
Another aim is to answer (ii),(iii), i.e., to construct a (generic) family of (−1)-minimal Q-curves E over a given quadratic field K, and family of curves C of genus two over Q which covers E. We shall also discuss the problem of constructing an algebraic correspondence on C defined over Q which induces the endomorphism√−2 on Jac(C).
One of the authors [6, 7] dealt with the same problems for the cases of (−1)-minimal Q-curves of degree 1 and 5.
2. STATEMENT OF RESULTS
We will give explicit equation of curve of genus two so that its Jacobian is isogenous to the Shimura abelian surface Af attached to a Hecke eigenform
whose Fourier coefficients fall into the field Q(√−2) of level N = 41, 24, 88, 152, and 344. Subsequently, we will construct a family of minimal Q-curves of degree 2 over quadratic fields.
Theorem 1. (1) There is a family of elliptic curves Ea,b: y2 = x3+ a
√
2x2+a
2 + 4b√2
4 x, a, b ∈ Q, which are(−1)-minimal Q-curves of degree 2 over Q(√2). (2) There is a family of elliptic curves with a, b ∈ Q
Ea,b: y2 = x3+ ax2+
a2+ 8bm
8 x, m
2
= −2, which are2-isogenous toσE
a,bover Q(
√
−2). Moreover, the curves Ea,bare(+1)-minimal Q-curves of degree 2.
(3) If N = α2− 2β2,α, β ∈ Q, then the elliptic curve E defined by
y2 = x3+1 − αm N ax2+ b − α(4N 2b + 4N bβ2 − a2β4) 4N2(N + 2β2) m x is a(+1)-minimal Q-curves over Q(√N ) of degree 2, where m2 = N , a, b ∈ Q.
Theorem 2. For a given positive square-free integer N , there exists a (−1)-minimal Q-curve of degree 2 over Q(√N ) if and only if N = α2+ 2β2 is a norm value of Q(√−2). In this case, each (−1)-minimal Q-curves over Q(
√
N ) of degree 2 is Q(√N )-isomorphic to one of the curve Ea,b: y2 = x3− 2a(α + m) β x 2 +αbN + β 2a2+ α2bm + β2bm α2 + β2 x
withm2 = N , where a, b, α, β are rational numbers.
In fact, we also have a similar result for the case of (+1)-minimal Q-curves of degree 2. That is, there exists a (+1)-minimal Q-curve of degree 2 over Q(√N ) if and only if N is a norm value of Q(√2).
Theorem 3. Let s be a rational number,
with H1(s) = − 2 + 8sx −2 + s2 + x 2 H2(s) = − 2 − 8(1 + s)(2 + s)x −2 + s2 + x 2 H3(s) = − 2 − 4(2 + 2s + s2)x −2 + s2 + x 2
is a family of curves over Q which covers (+1)-minimal Q-curves y2 = x3+6 3√−2t − 5 x−8 9√−2t − 7 , t = −2(s
2− 2)(s2+ 2s + 2)
(s2+ 4s + 2)2
over Q(√−2) of degree 2 so that End(Jac(C)) contains√2.
This is obtained by Hasegawa’s model [4] of generic family of Q-curves Ed,t(2) : y2 = x3+ 6(3mt − 5)x − 8(9mt − 7), with m2 = d of degree 2 over Q(m). In fact, the elliptic curve Ed,t(2) and its Galois
con-jugate are isogenous over Q(m,√−2). In the case of d = −2, E−2,t(2) is 2-isogenous over Q(√−2) to its conjugate.
Theorem 4. For a positive integer N > 2 with α2 + 2β2 = N = m2, the curve C : y2 = x5+ 8αx4+8α 2(3α2+ 5β2) α2+ β2 x 3 + 32αN x2+ 16N2x is of genus2 over Q, and covers the (−1)-minimal Q-curve
E : y2 = x3− 4(α2+ β2)(α + m)x2+ 4β4(α2+ β2)x
over Q(√N ) of degree 2 so that End(Jac(C)) contains Z[√−2].
The Q-curve E in this theorem is a special case of the family in 2 with parameters a = 2β(α2+ β2) and b = 0.
3. PRELIMINARIES
Most materials in this section can be found in many references, for ex-ample, [12, 13].
3.1. Elliptic Q-curves. An elliptic curve E defined over a number field K is called a Q-curve if for each σ ∈ Gal( ¯Q/Q), there exists an isogeny
φσ : E −→σE;
the curve E is of degree d if
d = min{deg(φσ) : φσ : E −→σE, id 6= σ ∈ Aut(K/Q)}.
In particular, if E is an elliptic Q-curve defined over a quadratic field K and the isogeny φ := φσ related to the non-trivial automorphism σ is defined
over K, then E is called minimal. Furthermore, we call E is ε-minimal Q-curve ifσφ ◦ φ = [εd], where ε = ±1.
3.2. Shimura abelian variety. For any positive integer N , consider the congruence subgroups Γ = Γ0(N ), or Γ1(N ) of SL(2, Z). Let Sk(Γ) be the
space of cusp forms of weight k with respect to the group Γ. With respect to the local parameter of the cusp ∞, a cusp form f has a q-Fourier expansion, namely
f (τ ) =X
n≥1
anqn, q = e2πiτ, Imτ > 0.
The so-called Hecke operators {Tn, n ≥ 1} defined on Sk(Γ1(N )) are
almost simultaneously diagonalizable with respect to the Petersson inner product. Therefore, we have common eigenvectors for Hecke operators. Furthermore, if the coefficient of q of a Hecke eigenform f is 1, we say that f is a normalized Hecke eigenform, i.e., Tnf = anf . In this situation, one
can show that the coefficients an are algebraic integers and the extension
field
Kf = Q(an : n > 1),
generated by all an, is a number field.
For each normalized common Hecke eigenform f of weight 2, Shimura associated to f an abelian variety Af over Q as a Q-simple of the Jacobian
of the modular curve X1(N ). The abelian variety Af is called the Shimura
abelian variety. Its dimension is [Kf : Q]. Moreover, under the action
of ap ∈ Kf corresponding to Hecke operator Tp, we can see that the
Q-algebra of endomorphisms of Af defined over Q is isomorphic to the field
Kf, namely, EndQAf ⊗ Q = Kf, and L(Af, s) = Y σ∈Gal( ¯Q/Q) L(σf, s),
where EndQ(Af) stands for the algebra of endomorphisms of Af defined
over Q, andσf =P
3.3. Modularity Theorem. In general, we say that an abelian variety A over Q is of GL2-typeif EndQA ⊗ Q contains a number field K with [K :
Q] = dim A. We remark that the abelian variety A/Q of GL2-type is simple
if and only if EndQA ⊗ Q = K is a number field and [K : Q] = dim A. In literature, we always say that the abelian variety A is of GL2-type under
the assumption that A is simple.
Trivially, an elliptic Q-curve and a Shimura abelian variety Af are of
GL2-type. As the modular parametrization of the elliptic curves defined
over Q, the Modularity Theorem also states that every abelian variety over Q of GL2-type is Q-isogenous to a Shimura abelian variety Af.
3.4. Shimura abelian surface attached to f with Nebentype character. Recall that the space S2(Γ1(N )) can be decomposed into the direct sum of
subsapeces
S2(Γ1(N )) =
M
χ
S2(N, χ),
where χ runs over all Dirichlet character mod N ; f ∈ S2(N, χ) satisfies
f aτ + b cτ + d = χ(d)(cτ + d)2f (τ ), ∀a b c d ∈ Γ0(N ).
In fact, for a normalized Hecke eigenform f of S2(N, χ), the field Kf is a
totally real field if χ is trivial, and Kf is a CM field for non-trivial χ. What
we are interested in is the Shimura abelian varieties attached to the Hecke eigenform with non-trivial character. We first consider the simpler cases which Kf is of degree 2 over Q and f is eigenform with the Kronecker
character N·.
3.5. Shimura abelian surfaces attached to f ∈ S2 N, N·. Let N be
the discriminant of the real quadratic field K = Q(√N ). Assume that there is a normalized Hecke eigenform f ∈ S2 N, N· such that Kf is
an imaginary quadratic field. Then the associated Shimura abelian surface Af is isogenous to the product of an elliptic curve Ef and its conjugateσEf
over K, where Ef is the so-called Shimura ’s elliptic curve, and σ is the
nonidentity element of Gal(K/Q). Moreover, Ef/K is a (−1)-minimal
Q-curve having good reduction everywhere, and its degree is the square-free part of discriminant D of the number field Kf.
In this report, we will focus on the related problems for the cases that f ∈ S2(N, N·) with Kf = Q(
√ −2).
4. EXAMPLES OFSHIMURAABELIANSURFACES
In this section, we use the notation EC to denote the elliptic curve
we use the computer algebra system Magma. For level N < 1000, the total number of N such that there exists f ∈ S2(N, N·) with Kf = Q(
√ −2) is 14. Let us show the examples for N = 24 and 41. For these examples, we will show that Jac(C) for the curve C of genus 2 over Q that we found is Q-isogenous to the related Shimura abelain surface.
4.1. N = 41. There is a normalized Hecke common eigenform f of S2 41, 41·
whose first few p-th Fourier coefficients are as follows
a2 a3 a5 a7 a11 a13 a17 a19
−1 −2√−2 2 2√−2 −2√−2 4√−2 0 −2√−2 The elliptic curve
EC : y2 = x x2+ 15 +√41 2 x + 33 + 5√41 2 !
obtained by Cremona [2] is a Shimura’s elliptic curve defined over K = Q(
√
41) associate to the attached Shimura abelian surface Af.
Proposition 1. The Jacobian variety of the hyperelliptic curve C : y2 = x(x2+ 62x + 1025)(x2− 2x + 1025)
is the Shimura abelian surface corresponding to the Hecke eigenformf . Proof. At first, we claim that
Jac(C) ∼
KEC × σE
C.
Observe that, there is an involution τ on C given by τ : (x, y) 7→ 1025 x , 5√41y x3 ! . It induces a 2-to-1 morphism φ from C to the elliptic curve
E : y2 = x(x + 62 − 10√41)(x − 2 − 10√41) given by φ : (x, y) 7−→ (x + 5 √ 41)2 x , y(x + 5√41) x2 ! .
Therefore, we also have a covering mapσφ from C toσE. From the map Φ = (φ,σφ), we can see that Jac(C) is isogenous to E×σE over the field K.
EC is 2-isogenous to the elliptic curve E over K. Precisely, the 2-isogeny from EC to E is given by (x, y) 7−→ x +15 + √ 41 2 + 33 + 5√41 2x , y 1 − 33 + 5√41 2x !! . By the characterization of the Weil restriction of a minimal Q-curve defined over a quadratic field, one has
Jac(C) ∼
QAf,
the Shimura abelian variety attached to the cusp form f . Remark 2. We can also compare the L-function of the curve C and that of f ,σf , to conclude that Jac(C) is the related Shimura abelian surface. Due to the work of Cremona [2], we have
L(E/Q(√41), s) = L(σE/Q(√41), s) = L(f, s)L(σf, s) and thus
L(C/Q(√41), s) = L(f, s)2L(σf, s)2.
Note that if prime p splits in Q(√41), then the Euler p-factor of L(f, s) and L(σf, s) are both equal to
1
1 − ap−s+ p1−2s, a ∈ Z;
the Euler p-factor of L(C/Q(√41)) is 1/(1 − ap−s+ p1−2s)4. If p is inert
in Q(√41), then the Euler p-factors of L(f, s) and L(σf, s) are
1
1 ± b√−2p−s+ p1−2s, b ∈ Z;
the Euler p-factors of L(C/Q(√41)) is 1
(1 − α2p−2s)2(1 − β2p−2s)2,
where α = (b√−2 +p−2b2− 4p)/2 and β = (b√−2 −p−2b2− 4p)/2.
On the other hand, for each prime p of good reduction, write the Euler p-factor of L(C/Q, s) as
Lp =
1
(1 − c1p−s)(1 − c2p−s)(1 − c3p−s)(1 − c4p−s)
, for some constants ci. If p splits in Q(
√
41), then the Euler p-factor of L(C/Q(√41), s) is L2p. Therefore, the Euler p-factor of L(C/Q, s) must be
Lp =
1
(1 − ap−s+ p1−2s)2 = the Euler p-factor of L(f, s)L( σ
If p is inert in Q(√41), then the Euler p-factor of L(C/Q(√41), s) is 1
(1 − c2
1p−2s)(1 − c22p−2s)(1 − c23p−2s)(1 − c24p−2s)
. We can find that
c21 = c23 = α2, c22 = c24 = β2.
Now observe that C has the involution (x, y) 7→ ((5√41)2/x, (5√41)3y/x3),
thus we must have c1+ c2+ c3+ c4 = 0. Therefore,
{c1, c2c3, c4} = {±(b
√
−2 ±p−2b2− 4p)/2},
and the Euler p-factor of L(C/Q, s) must be
Lp = the Euler p-factor of L(f, s)L(σf, s).
Hence for almost all prime p, the the Euler p-factor of L(C/Q, s) is equal to the Euler p-factor of L(f, s)L(σf, s).
From the property of a Shimura abelian variety, we know that EndQ(Jac(C)) contains√−2. Here, we give a way to verify that√−2 acts on the Jacobian of C.
Lemma 3. The Q-algebra of endomorphisms of Jac(C) contains√−2. Proof. To show this, we will find an algebraic correspondence (equation) T ⊂ C × C which induces the endomorphism φ on Jac(C) defined over Q so that φ2 = [−2].
Identify Jac(C) as Pic0(C), the equivalence class of divisors on C of degree 0. Let T be the set
T = {((x, y), (u, w)) ∈ C × C : AQ(x, u) = 0, BQ(w, y, x, u) = 0}, where AQ(x, u) = 25(205 + 3x)2+ u2(15 + x)2+ 6u(5125 + 246x + 5x2) and BQ(w, y, x, u) =5 · 212wx + (205 + 3u)(−1125 + 192u + 5u2)y + (u − 15)(u + 15)2xy.
Then for a given point P = (u, v) on C, we have exactly two points Q1 = (x1, y1), Q2 = (x2, y2) on C such that (P, Q1), (P, Q2) ∈ T. This
determines a map T on Pic0(C) defined by
(u, v) 7→ (x1, y1) + (x2, y2).
We recall that each endomorphism of Jac(C) induces an endomorphism of the space of holomorphic differential 1-forms on C, H0(C, Ω1). Then,
under the action of the induced map associate to T, the holomorphic 1-forms dx/y, xdx/y become
ω1 = du w 7→ ω ∗ 1 = dx1 y1 + dx2 y2 ω2 = udu w 7→ ω ∗ 2 = x1dx1 y1 +x2dx2 y2 . Write AQ(x, u) = A2(u)x2+ A1(u)x + A0(u) and let
W (u) =x1+ x2 = −A1(u)/A2(u)
P (u) =x1x2 = A0(u)/A2(u).
Then W (u) = −30u 2+ 1476u + 30750 u2+ 30u + 225 P (u) = 225u2+ 30750u + 1050625 u2+ 30u + 225 ,
and we can express ω1∗and ω∗2 as ω∗1 =−3(205 + 3u)(u − 15) 320(u + 15) du w + 25(−1125 + 192u + 5u 2)(205 + 3u)2 64P (u)(u + 15)3 du w = 3 4 + u 20 du w, ω∗2 = −3(205 + 3u)(u − 15)W (u) 320(u + 15) du w − (205 + 3u)(85u 3+ 3339u2+ 39915u − 556875) 160(u + 15)3 du w = − 205 4 + 3 4u du w.
Therefore, the endomorphism induced from T on H0(C, Ω1) is defined
over Q and T2 = [−2], which implies that EndQ(Jac(C)) contains√−2. 4.2. N = 24.
Proposition 4. Let {f,σf } be a basis of normalized newforms of S2 24, 24·
with the first fewp-th Fourier coefficients of f are
a2 a3 a5 a7 a11 a13 a17 a19
−a a − 1 0 0 −2a 0 4a 2
witha =√−2. Then the hyperelliptic curve
over Q is the Shimura abelian suface corresponding to the eigenform f . The curve C covers the elliptic curve
E : Y2 = 22 − 9 √ 6 4 X2+ (30 + 12√6)X + 20√6 + 49 X − 5 − 2√6 and its conjugateσE, where
X = x + √ 6 x −√6 !2 , Y = 4y (x −√6)3.
The curve E is isomorphic to the elliptic curve EC : y2+
√
6xy+1 +√6y = x3+1 −√6x2−3√6 + 1x−1−2√6 obtained by Cremona via the map
X = −4(√6 + 2)x − (7 + 4√6), Y = 8y + 4√6x + 4(1 +√6). Therefore, we can see that the Jac(C) is isogenous over Q(√6) to the EC× σE
C and thus Jac(C) is the Shimura abelian variety attached to the cusp f .
Lemma 5. The endomorphism ring End(Jac(C)) contains√−2. Proof. The algebraic correspondence given by the equations
A(x, u) = 0, B(w, y, x, u) = 0 with A(x, u) = u2(12 − 6x + x2) − 6u(6 − 4x + x2) + 12(3 − 3x + x2) and B(w, y, x, u) =36w(36 − 24u + u3− 24x + 18ux − 3u2x) − (6 − 6u + u2)(12 − 6u + u2)2y,
induces an endomorphism T defined over Q with T2 = −2 · id. To verify
this, we find that the representation matrix of T ∈ End(H0(C, Ω1)) is
−2 1 −6 2
, with respect to the basis {dx/y, xdx/y}.
More precisely, for a given point (u, w) on C, we define T : H0(C, Ω1) −→ H0(C, Ω1) by du w 7→ dx1 y1 + dx2 y2 udu w 7→ x1dx1 y1 +x2dx2 y2 ,
where ((u, w), (x1, y1)), ((u, w), (x2, y2)) are elements of the set
T = {((x, y), (u, w)) ∈ C × C : A(x, u) = 0, B(w, y, x, u) = 0}. For the fixed point (u, v), one has
W (u) =x1+ x2 = 18 − 12u + 3u2 3 − 3u + u2 P (u) =x1x2 = 36 − 18u + 3u2 3 − 3u + u2 .
By direct computation, we find that dx1 y1 +dx2 y2 =(u − 2)du w, x1dx1 y1 + x2dx2 y2 =(2u − 6)du w. 4.3. Key idea to construct the associate Shimura abelain surface. Our idea to find curves C in previous examples is as follows.
Note that if a curve C defined over K of genus 2 has a double cover of some elliptic curve, equivalently, it has a non-hyperelliptic involution. An involution on C is of the form
τ : (x, y) 7→ ax + b cx − a, (a2+ bc)3/2y (cx − a)3 ,
if C is given by the equation y2 = F (x), where deg F (x) = 5 or 6 and F (x) ∈ K[x] has no repeat roots, and (a b
c −a) ∈ GL(2, K). In our cases, we
assume the involution is of the form τ : (x, y) 7→ a 2N x , a√N y x3 ! , a ∈ Q.
Thus, if the equation of the quotient elliptic curve defined over Q(√N ) is given, together with the action of the involution τ , it will give us a chance to find the curve C : y2 = F (x) with F (x) ∈ Q[x].
In the case of level 41, we have the equation of Shimura elliptic curves E : Y2 = G(X) and σE defined over Q(√41). We suppose that the
relations between x, y and X and Y are X = (x + u) 2 x , Y = y(x + u) x2 , u ∈ Q( √ 41). This is invariant under the involution (x, y) 7→ (u2/x, u3y/x3).
For level 24, actually, we are not so lucky to find the curve C as that in the case of N = 41. In this case, we can not find a genus 2 lift which is defined
over Q using the same argument as in level 41. Here we consider the other way that C covers the elliptic E which is quotient by the involution (x, y) 7→ (−x, y), and coversσE quotient by (x, y) 7→ (−x, −y). Roughly speaking,
we consider some suitable quadratic twist u of the given Shimura’s elliptic curve (e.g. u = 2 −√6). Then we find genus 2 curve defined over Q(√6), say C0. Again, we twist the curve C0 by u, we can get a curve C which is lift from E. By changing coordinate, we find the genus 2 curve C defined over Q covering E andσE.
⊗u
C0 −→ C00 −→ C
↓
E0 ←− E ⊗u
For the case of level 88, 152, and 344, we use similar ways to find the curves C (Please see the section Appendix). However, in general, we are not always lucky. For example, for N = 337 and N = 881, there still have some difficulties to get genus 2 defined over Q successfully.
5. PROOFS OFTHEOREMS
5.1. Construction of family of Q-curves of degree 2. The main idea for construction of degree 2 Q-curves is using the following known isogeny between elliptic curves. Let E be an elliptic curve y2 = x3 + Ax2 + Bx
with defined over K with B(A2− 4B) 6= 0. Then there is an K-isogeny of
degree 2 from E to the elliptic curve
E0 : y2 = x3− 2Ax2+ (A2− 4B)x by φ : (x, y) 7→ y 2 x2, y(B − x2) x2 , and the dual isogeny of φ is
b φ : (x, y) 7→ y 2 4x2, y (A2− 4B − x2) 8x2 . Proof of Theorem 1.
Proof. First of all, let us consider the case of the family of Q-curves Ea,bin Theorem 1 over the field Q(√2). For a fixed elliptic curve
E = Ea,b: y2 = x3+ amx2+
a2+ 4bm
4 x, m
its conjugateσE
σE : y2 = x3− amx2+ a2− 4bm
4 x
is isogenous to the curve
σE0
: y2 = x3+ 2amx2+ (4bm + a2)x
over Q(√2) of degree 2. Moreover, E is isomorphic to the curveσE0 over
Q( √
2) given by the map
σ
E0 −→ E, (x, y) 7→ (x/m2, y/m3) = (x/2, y/2m). Therefore, we have an isogeny φ : E −→σE over Q(√2) of degree 2
φ : (x, y) 7→ y 2 2x2, ym(a2 + 4bm − 4x2) 16x2 , Observe that the dual isogeny of φ is
b φ : (x, y) 7→ y 2 2x2, y(a2− 4bm − 4x2) 8mx2 , and the conjugate of φ is
σφ : (x, y) 7→ y 2 2x2, − ym(a2− 4bm − 4x2) 16x2 , so we have bφ = −σφ.
Likewise, we can verify that the respect families of elliptic Q-curves de-scribed in Theorem 1 are (±1)-minimal Q-curves of degree 2 over the
re-spect quadratic fields.
Proof of Theorem 2.
Proof. Let N be an integer greater than 2. We now suppose that α, β are two rational numbers so that N is equal to α2+ 2β2. For any rational numbers a and b, the elliptic curve E defined by the equation
y2 = x3−2a(α + m)
β x
2+ αbN + β2a2+ α2bm + β2bm
α2+ β2 x
is an elliptic curve over Q(√N ) with m2 = N . It is isomorphic to the curve
σE0
: y2 = x3− 4a(α − m)
β x
2 +4(αbN + β2a2− α2bm − β2bm)
α2+ β2 x
over Q(√N ) via the map
σE0 −→ E,
Therefore, we have an isogeny φ : E −→σE over Q(√N ) of degree 2 φ(x, y) = y2(m − α)2 4β2x2 , y(αbN + β2a2− α2bm − β2bm + x2β2− N x2)β3 x2(N − β2)(−α + m)3 ; the dual isogeny of φ is
b φ(x, y) = y2β2 (m − α)2x2, y(αbN + β2a2+ α2bm + β2bm + x2β2− N x2)β3 x2(N − β2)(α + m)3 .
Therefore, we have bφ = −σφ and thus E is a (-1)-minimal Q-curve of degree 2 defined over Q(√N ).
For the converse, we assume that E is a minimal Q-curve of degree 2 over the field real quadratic field K = Q(√N ). Hence we can consider thatσE as E/ < kerφ > for some K-isogeny with |kerφ| = 2. By suitable
changing coordinate, we may assume E is of the form y2 = x(x2+Ax+B)
and σE is isomorphic to the curve E0 : y2 = x(x2 − 2Ax + A2 − 4B),
where A, B ∈ K. Therefore, there exist some number u ∈ K∗ so that σ(A)u2 = −2A, σ(B)u4 = A2 − 2B. Moreover, the curve E is (−1)-minimal, so we have the relationσφ = − ˆφ and hence
u2σ(u2) = 1/4, and 8u3σ(u3) σ(B) − x2 = x2+4σ(Bu4)−σ(A2u4). This implies that uσ(u) = −1/2, i.e. there exist an element a + b√N in Q(
√
N ) with norm −2. Equivalently, the integer N must be of the form α2+ 2β2
for some α, β ∈ Q.
5.2. Genus 2 lifts of certain Q-curves. In this note, we only construct some special cases of genus 2 lifts form certain Q-curves of degree 2. For the further study, we will try to find more general equations of the hyperel-liptic curves that lift from the families of Q-curves of degree 2. In this sec-tion, we will give the equations of algebraic correspondences for the curves C in Theorem 4, 3 that determine the endomorphismsñ2 on Jac(C).
For the curve in Theorem 3 C : y2 = −2 + 8sx −2 + s2 + x 2 −2 −8(1 + s)(2 + s)x −2 + s2 + x 2 −2 − 4(2 + 2s + s 2)x −2 + s2 + x 2 ,
the algebraic correspondence defined by the equations A(x, u) =(6 + 4s + s2)(2 + 4s + 3s2) + 2(−2 + s2)(2 + 2s + s2)u − (−2 + s2)2u2/2 + 2(−2 + s2)(2 + 2s + s2)x + 2(−2 + s2)2xu − (−2 + s2)(2 + 2s + s2)xu2 − (−2 + s2)2x2/2 − (−2 + s2)(2 + 2s + s2)x2u + (6 + 4s + s2)(2 + 4s + 3s2)x2u2/4 B(w, y, x, u) = − 8(2 + 4s + s2)2(4 − 2s2+ 8su − 2u2+ s2u2) (4 − 2s2− 16u − 24su − 8s2u − 2u2+ s2u2) (32 + 32s − 16s2− 32s3− 8s4+ 8s5+ 4s6− 16u + 24s2u − 12s4u + 2s6u − 80u2 − 208su2− 280s2u2− 240s3u2 − 140s4u2− 52s5u2− 10s6u2− 8u3+ 12s2u3− 6s4u3 + s6u3 − 16x + 24s2x − 12s4x + 2s6x − 16ux − 16sux + 8s2ux + 16s3ux + 4s4ux − 4s5ux − 2s6ux + 88u2x + 128su2x + 60s2u2x − 30s4u2x − 32s5u2x − 11s6u2x + 40u3x + 104su3x + 140s2u3x + 120s3u3x + 70s4u3x + 26s5u3x + 5s6u3x)
+ (−2 + s2)3(−8 + 8s2− 2s4+ 16u + 16su − 8s3u − 4s4u + 12u2 + 32su2+ 36s2u2 + 16s3u2+ 3s4u2)2wy
gives us an automorphism T as√2 on Jac(C). Regarding Jac(C) as Pic0(C), we let
T := {((u, w), (x, y)) ∈ C × C : A(x, u) = 0, B(w, y, x, u) = 0}, Then for each (u, w) on C, we have exact two points (x1, y1) and (x2, y2)
such that ((u, w), (x1, y1)) and ((u, w), (x2, y2)) in T . We then define T in
the rule
(u, w) 7→ (x1, y1) + (x2, y2).
This induces an endomorphism T defined over Q with T2 = [2] on the space H0(C, Ω1) given by dx y = xdx y and xdx y = 2 dx y . For the curve
C : y2 = x5+ 8ax4+ 8a
2(3a2+ 5b2)
a2+ b2 x 3
N = m2 with N > 2 in 4, we can determine a map T defined over Q gives us√−2 on Jac(C) from the algebraic correspondence
A(x, u) =(a2+ b2)N2u2(a + 2x)2+ 4(a2+ b2)N2(N + 2ax)2
+ 4aN2u(a4+ 3a2b2+ 2b4+ 4a3x + 8ab2x + 4a2x2 + 4b2x2) B(w, y, x, u) =2bx2(16a6+ 80a4b2+ 128a2b4+ 64b6+ 32a5u
+ 96a3b2u + 64ab4u + 24a4u2+ 40a2b2u2 + 8a3u3+ 8ab2u3+ a2u4+ b2u4)
(4a6+ 20a4b2+ 32a2b4+ 16b6+ 8a3b2u + 16ab4u − a4u2− a2b2u2 + 8a5x + 24a3b2x
+ 16ab4x + 8a2b2ux + 8b4ux − 2a3u2x − 2ab2u2x) − wy(a2+ b2)2(2a2+ 4b2+ au)3.
6. MODULAR PARAMETRIZATION
As the modular parametrization of rational elliptic curves, we naturally expect that there is a modular parametrization of the genus 2 lifts whose Jacobian variety is of GL2-type. Unfortunately, it may be impossible to find
modular parametrization of such curves. In this section, we will give an ex-ample to show that it is possible to find a parametrization of a genus 2 curve which is lifting from a (+1)-minimal Q-curve, and give another example to see the it is also possible that we can not find any parametrization of a curve of genus 2 which covers a (−1)-minimal elliptic Q-curves.
6.1. Subvariety of Jac(X0(256)). Hasegawa [4] showed that the
hyper-elliptic curve
C : y2 = x(x4− 64)
covers a (+1)-minimal Q-curve of degree 2 defined over Q(√−1). Its Ja-cobian variety is of GL2-type with
√
2 multiplication, and its L-function is equal to the product L(f, s)L(σf, s), where f is the newform
f = q +√2q3+ 5q9−√2q11+ 6q17− 3√2q19+ . . . . on Γ0(256). If we let f1 =(f + fσ)/2 = q + 5q9+ 6q17− 5q25− 8q33+ 6q41+ . . . , f2 =(f − fσ)/2 √ 2 = q3− q11− 3q19+ 2q27+ 3q43+ 6q51+ . . . , then we can write x and y as modular functions
x = f1/f2 and y = −
qdx/dq 2f2
on Γ0(256). Moreover, we can construct the functions x and y explicitly.
To be more precise, from the work of Tu and Yang [14], we can express x and y as x = (s2− 4s + 4)/t2 and y = −t 3(−24s3+ 48s2+ 22s4+ s5 − 176s − 32) −t8+ 896s3+ 512s4+ 8s7+ 512s + 1024s2 + 224s5+ 64s6 where s = η(128τ ) 6 η(64τ )2η(256τ )4 and t = η(16τ )2η(128τ ) η(8τ )η(256τ )2
are generators of the function field of X0(256). This gives us a modular
parametrization of the curve C.
6.2. Shimura abelian surface attached to cusp form of level 24. From the example in section 4.2, the Jacobian Jac(C) of the hyperelliptic curve
C : y2 = (x2− 6x + 6)(x4− 6x3+ 18x2− 36x + 36)
is the Shimura abelian surface corresponding to the normalized newforms f andσf with character 24· and Kf = Q(
√
−2). In this case, we will give a brief explanation for the nonexistence of modular parametrization from functions on X1(24).
Let Γ be the subgroup between Γ0(24) and Γ1(24) corresponding to the
character 24· . Then the modular curve X(Γ) obtained by Γ is of genus 3 and non-hyperelliptic. The cusp forms corresponding to the curve C are invariant under the action of 24· , so if there is a modular parameterization, then the modular functions should also be invariant under Γ. Thus, if there is a morphism from X1(24) to C, there should also be a morphism from
X(Γ) to C. However, according to the following lemma, it tells us that there does not exist a morphism from X(Γ) to the curve C.
Lemma 6. Suppose that X and Y are algebraic curves of genus 3 and 2, re-spectively. If there is a finite morphismX −→ Y , then X is a hyperelliptic curve.
So, it is natural to address the question:
How to describe the ”modular parametrization” of the curve C?
It is an interesting problem to find a way or formula to state the modular parametrization of the genus 2 lifts from Q-curves whose Jacobians are of GL2-type.
7. APPENDIX
Here, we list the cases of N = 88, 152 and 344. All of these cases, the elliptic curves E : Y2 = f (X) are isomorphic to the elliptic curves E
K
obtained by Kida [10], and the maps from the curves C : y2 = F (x) of genus 2 to E are X = x + √ N x −√N !2 , Y = y (x −√N )3.
In the following, we will list the curves C over Q of genus 2 whose Jacobian is Shimura abelian variety attached to the pair of the normalized newforms f ,σf in S
2 N, N· with Kf = Q(
√
−2). Also, we give an algebraic cor-respondence which defines an endomorphism T defined over Q on Jac(C) such that T2 = [−2]. N = 88 C : y2 = (7x2− 66x + 154)(25x4− 462x3+ 3234x2− 10164x + 12100) E : Y2 = X3− 376320(−294 + 63√22)2X − 78675968(−294 + 63√22)3 A(x, u) = − 77u(66 − 28x + 3x2) + u2(539 − 231x + 25x2) + 11(1100 − 462x + 49x2) B(w, y, x, u) = − wy(1100 − 462u + 49u2)2
+ 2x2(12100 − 10164u + 3234u2− 462u3 + 25u4)
(−71632 + 30492u − 3234u2+ 10164x − 3256ux + 49u3x) The algebraic correspondence A(x, u) = 0, B(w, y, x, u) = 0 leads the result T =−14−66 143 ∈ End(H0(C, Ω1)) ,→ End QJac(C), T2 = [−2]. N = 152 C : y2 = 2(3x2− 38x + 114)(5x4− 114x3+ 1026x2− 4332x + 7220) E : Y2 = X3− 376320(−126 + 21√38)2x − 78675968(−126 + 21√38)3 A(x, u) = − 57u(38 − 12x + x2) + u2(171 − 57x + 5x2) + 19(380 − 114x + 9x2)
B(w, y, x, u) =2x2(7220 − 4332u + 1026u2− 114u3+ 5u4)
(−40432 + 12996u − 1026u2+ 4332x − 1064ux + 9u3x) − wy(380 − 114u + 9u2)2
T = −6−38 61 ∈ End(H0(C, Ω1)), T2 = [−2]. N = 344 C : y2 =2(51x2− 946x + 4386) (1301x4 − 48246x3+ 671058x2− 4149156x + 9622196) E : Y2 =X3− 376320(−2142 + 231√86)2X − 78675968(−2142 + 231√86)3 A(x, u) =2(51x2− 946x + 4386) (1301x4− 48246x3+ 671058x2− 4149156x + 9622196)
B(w, y, x, u) =2x2(9622196 − 4149156u + 671058u2− 48246u3
+ 1301u4)(−57718384 + 12447468u − 671058u2 + 4149156x − 671144ux + 2601u3x)
− wy(223772 − 48246u + 2601u2)2
T =−102−946 10211
∈ End(H0(C, Ω1)), T2 = [−2].
REFERENCES
[1] G. Cardona, J. Gonz´alez, J. C. Lario, and A. Rio. On curves of genus 2 with Jacobian of GL2-type. Manuscripta Math., 98(1):37–54, 1999.
[2] J. E. Cremona. Modular symbols for Γ1(N ) and elliptic curves with everywhere good reduction. Math. Proc. Cambridge Philos. Soc., 111(2):199–218, 1992.
[3] Jordan S. Ellenberg and Chris Skinner. On the modularity of Q-curves. Duke Math. J., 109(1):97–122, 2001.
[4] Yuji Hasegawa. Q-curves over quadratic fields. Manuscripta Math., 94(3):347–364, 1997.
[5] Yuji Hasegawa, Ki-Ichiro Hashimoto, and Fumiyuki Momose. Modularity conjecture for Q-curves and QM-curves. Internat. J. Math., 10(8):1011–1036, 1999.
[6] Ki-ichiro Hashimoto. Q-curves of degree 5 and Jacobian surfaces of GL2-type. Manuscripta Math., 98(2):165–182, 1999.
[7] Ki-ichiro Hashimoto. Q-curves with rational j-invariants and Jacobian surfaces of GL2-type. In Galois theory and modular forms, volume 11 of Dev. Math., pages 39– 64. Kluwer Acad. Publ., Boston, MA, 2004.
[8] Haruzo Hida. Geometric modular forms and elliptic curves. World Scientific Pub-lishing Co. Inc., River Edge, NJ, 2000.
[9] Chandrashekhar Khare and Jean-Pierre Wintenberger. Serre’s modularity conjecture. I. Invent. Math., 178(3):485–504, 2009.
[10] Masanari Kida. Computing elliptic curves having good reduction everywhere over quadratic fields. Tokyo J. Math., 24(2):545–558, 2001.
[11] Kenneth A. Ribet. Abelian varieties over Q and modular forms. In Algebra and topol-ogy 1992 (Taej˘on), pages 53–79. Korea Adv. Inst. Sci. Tech., Taej˘on, 1992.
[12] Kenneth A. Ribet. Abelian varieties over Q and modular forms. In Modular curves and abelian varieties, volume 224 of Progr. Math., pages 241–261. Birkh¨auser, Basel, 2004.
[13] Goro Shimura. Introduction to the arithmetic theory of automorphic functions, vol-ume 11 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original, Kanˆo Memorial Lectures, 1. [14] Fang-Ting Tu and Yifan Yang. Defining equations of X0(22n). Osaka J. Math.,
46(1):105–113, 2009.
DEPARTMENT OFMATHEMATICS, WASEDAUNIVERSITY, 3-4-1, OKUBOSHINJUKU
-KU, TOKYO, 169-8555 JAPAN
E-mail address: khasimot@mse.waseda.ac.jp
DEPARTMENT OFMATHEMATICS, WASEDAUNIVERSITY, 3-4-1, OKUBOSHINJUKU
-KU, TOKYO, 169-8555 JAPAN
