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Examples of abelian surfaces with everywhere good reduction

We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler-Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms with paramodular level structures.


Introduction
A celebrated result of Fontaine [20] (see also Abrashkin [1]) asserts that there is no abelian scheme over Z. In other words, there is no abelian variety over Q with everywhere good reduction. However, long before this result, there were a few examples of elliptic curves of unit conductor over quadratic fields in the literature. For example, the curve E : y 2 + xy + 2 y = x 3 , where = 5+ √ 29 2 is the fundamental unit in F = Q( √ 29), was known to Tate, and to Serre who extensively studied it in [38]; it is also alluded to in [42]. Since then, there has been much work on finding elliptic curves with everywhere good reduction over number fields, with a particular emphasis on quadratic fields; see for example [40,47,12,8,36,28]. For real quadratic fields, the database of such curves has been considerably expanded by Elkies [16]. In [14], it is shown that this database is complete for all fundamental discriminants ≤ 1000 of narrow class number one, if one assumes modularity. A more systematic algorithm which, given a number field F and a finite set of primes S of its ring of integers, returns the set of all elliptic curves over F with good reduction outside of S is given in [11]. However, this algorithm relies on algorithms for S-integral points for elliptic curves, and has not yet been full implemented for this reason. An alternate (and perhaps more efficient) approach which uses S-unit equations is currently being explored by Cremona and Elkies [10]. In fact, a similar method has already been used by Smart [45] to find hyperelliptic curves of genus 2 with good reduction outside S when F = Q.
Date: October 20, 2013. AK was supported in part by National Science Foundation grant DMS-0952486 and by a grant from the Solomon Buchsbaum Research Fund. LD is supported by an EPSRC Career Acceleration Fellowship.
In contrast, and to the best of our knowledge, there is not a single example of an abelian surface with everywhere good reduction in the literature (except in the case of complex multiplication [13], or when the abelian surface is a product of elliptic curves). This could possibly be explained by the fact that all the algorithms we mentioned above do not readily generalize to the genus 2 situation. The goal of this paper is to remedy that situation by providing the first equations for such surfaces over real quadratic fields.
We note that the non-existence of abelian varieties over Q with good reduction everywhere is instrumental to the Khare-Wintenberger proof of the Serre conjecture for Galois representations of Gal(Q/Q). As described in [29], the proof of the Serre conjecture in retrospect can be viewed as a method to exploit an accident which occurs in three different guises: (a) (Fontaine, Abrashkin) There are no non-zero abelian varieties over Z. (b) (Serre, Tate) There are no irreducible representations where F is the algebraic closure of F 2 or F 3 that are unramified outside of 2 and 3 respectively. (c) S 2 (SL 2 (Z)) = 0, i.e., there are no cusp forms of level SL 2 (Z) and weight 2.
The failure of this happy accident over general number fields, such as real quadratic fields, means that new techniques are needed for analogous modularity results. Our approach to the construction of abelian surfaces with everywhere good reduction combines three key elements: (a) recent advances in the computation of Hecke eigenvalues of Hilbert modular forms, (b) new rational models of Hilbert modular surfaces, and (c) the Eichler-Shimura conjecture for Hilbert modular forms. As a result of our investigation, we produce further evidence for the Eichler-Shimura conjecture, as well as for a conjecture of Brumer and Kramer [6] associating abelian varieties to paramodular Siegel modular forms on Sp (4).
The outline of the paper is as follows: in Section 2, we briefly recall the basic facts regarding these three ingredients. In Section 3, we describe our strategy to predict and find examples of good reduction abelian surfaces, assuming the Eichler-Shimura conjecture. Section 4 provides several illustrative examples of our methods, giving explicit abelian surfaces with good reduction everywhere, and connecting them to appropriate Hilbert modular forms. We conclude with a list of all our examples in Section 5.
Let k ≥ 2 be an even integer. A Hilbert modular form of (parallel) weight k and level N is a holomorphic function f : Let f be a Hilbert modular form of weight k and level N.
Hence, by the Koecher principle [5], f admits a q-expansion of the form We say that f is a cusp form if a 0 = 0. Let f be a cusp form of weight k and level N, and µ ∈ F × a totally positive. Then since f is invariant under the image of O × F in SL 2 (O F ) which acts as z → 2 z, ∈ O × F , the quantity a 2 µ is independent of ∈ O × F . So, for any ideal m ⊆ O F , the quantity a m (f ) = a µ , where µ is a totally positive generator of m, is well-defined and depends only on m. We call it the m-th Fourier coefficient of f . When f is a normalized eigenform for the Hecke operators (i.e. a (1) (f ) = 1), the eigenvalue of the Hecke operator T m is a m (f ) for each m N. It is a theorem of Shimura [44] that in this situation, the a m (f ) are algebraic integers and the Z-subalgebra has finite rank and is therefore an order in some number field K f (called the field of Fourier coefficients of f ). For more background on Hilbert modular forms, see [14,19,5].
Here, we wish to point out some new techniques in the computation of Hilbert modular forms, which arise from the Eichler-Jacquet-Langlands-Shimizu correspondence between Hilbert modular forms and quaternionic modular forms. We will not go into details here, but instead refer the reader to [14] for a detailed description of these methods. The upshot is that it is possible to efficiently compute systems of Hecke eigenvalues for Hilbert modular cusp forms by instead computing modular forms on finite spaces or on Shimura curves. This will be crucial to our methods in this paper. The corresponding algorithms have been implemented in the Hilbert Modular Forms Package in Magma [4]).
2.2. Hilbert modular surfaces. Let K be a real quadratic field of discriminant D . The Hilbert modular surface Y − (D ) is a compactification of the coarse moduli space which parametrizes principally polarized abelian surfaces with real multiplication by the ring of integers by adding finitely many cusps, where H + and H − are the upper and lower half-planes respectively. In the 1970s, Hirzebruch and others analyzed the geometric invariants of these surfaces and determined their Enriques-Kodaira classification. The Hilbert modular surface maps to the A 2 , the moduli space of principally polarized abelian surfaces, by forgetting the action of O K . Its image is the Humbert surface H D , and the map Y − (D ) → H D is a double cover, ramified along a union of modular curves. The surfaces Y − (D ) have models over the integers, with good reduction away from primes dividing D .
Recently, Elkies and the second author [17] computed explicit birational models over Q for these Hilbert modular surfaces for all the fundamental discriminants D less than 100, by identifying the Humbert surface H D with a moduli space of elliptic K3 surfaces, which may be computed explicitly. For the fundamental discriminants in the range 1 < D < 100, the Humbert surface is a rational surface, i.e. birational to P 2 over Q (and in fact, even over Q). Therefore, they are able to exhibit H D as a double cover of P 2 , with equation z 2 = f (r, s), where r, s are parameters on P 2 . They also get the map to A 2 , which is birational to M 2 , the moduli space of genus 2 curves. It is given by producing the Igusa-Clebsch invariants of the image point as rational functions of r and s. where When F = Q, this conjecture is a theorem, due to Eichler for prime level and Shimura in the general case. The Eichler-Shimura construction can be summarized as follows. Let N > 1 be an integer, and let X 1 (N ) be the modular curve of level Γ 1 (N ). This curve and its Jacobian J 1 (N ) are defined over Q. We recall that the space S 2 (Γ 1 (N )) of cusp forms of weight 2 and level Γ 1 (N ) is a T-module, where T is the Hecke algebra. Let f ∈ S 2 (Γ 1 (N )) be a newform, and let I f = Ann T (f ). Shimura [43] showed that the quotient is an abelian variety A f of dimension [K f : Q] defined over Q with endomorphisms by the order O f = Z[a n (f ) : n ≥ 1] and that where [f ] denotes the Galois orbit of f .
One of the main consequences of the proof of the Serre conjecture [39] by Khare-Wintenberger [30] is that the converse to Conjecture 1 is true when F = Q. That is, an abelian variety of GL 2 -type is isogenous to a Q-simple factor of J 1 (N ) for some N [31]. And so, this provides a theoretical construction of all abelian varieties of GL 2 -type over Q with a prescribed conductor. In fact, one can make this explicit in many cases (see [9] for elliptic curves, and [23,25] for abelian surfaces).
For [F : Q] > 1, the known cases of Conjecture 1 exploit the cohomology of Shimura curves. For instance, the conjecture is known when [F : Q] is odd, or when N is exactly divisible by a prime p of O F [48]. The simplest case in which Conjecture 1 is still unknown is when f is a newform of level (1) and parallel weight 2 over a real quadratic field. In that case, the conjecture predicts that the associated abelian variety A f has everywhere good reduction.

The strategy
Let F be a number field of class number 1, and E an elliptic curve over F given by a (global minimal) Weierstrass equation Since E has everywhere good reduction if and only if ∆ is a unit in O F , we can find all the elliptic curves over F with everywhere good reduction by solving (1) as ∆ runs over a finite set of representatives of O × F /(O × F ) 12 . Most of the algorithms we mentioned earlier rely on this fact.
Unfortunately, abelian varieties of higher dimension are not characterized by a nice single diophantine equation as (1). For this reason, we need an additional ingredient which will guide our search. This extra input is provided by the Eichler-Shimura conjecture.
Suppose we have a Hilbert modular eigenform f of weight 2 over F , with Hecke eigenvalues a m (f ) in a real quadratic field K f of discriminant D . The Eichler-Shimura conjecture predicts that there should be an abelian variety A over F of dimension [K f : Q] = 2, (up to isogeny) associated to this data, which has real multiplication by an order in K f . Furthermore, the conductor of A should divide the level N of f . In particular, if f has level (1), the conjectural abelian surface A has good reduction everywhere. This observation will be the source of our examples in this paper, for which the abelian surface turns out to be principally polarized, and also has real multiplication by the full ring of integers of K f . Our strategy to produce such A is as follows: (a) Find a Hilbert modular form of level (1) and weight 2 for a real quadratic field F , with coefficients in a real quadratic field K f of discriminant D . Note that there is no reason one has to restrict to the case when the base field is a real quadratic field F . The next interesting case in which the Eichler-Shimura conjecture is not known is that of totally real quartic base fields L. So one could look for eigenforms of weight 2 for SL 2 (O L ) whose Fourier coefficients are in a real quadratic field K of discriminant D, and on the other hand try to find L-rational points on the Hilbert modular surface Y − (D). In this paper, we looked at quadratic base fields F for convenience. On the other hand, if we instead want examples for which the field K f has larger degree, we might need explicit rational models for the appropriate Hilbert modular varieties, which are not currently available. Hence the choice of K is restricted.
For simplicity, we investigated only real quadratic fields F of narrow class number 1 and discriminant less than 1000. We found twenty-eight examples of Hilbert newforms, and corresponding abelian surfaces for most of these forms. We will say a few words later about the "missing" examples, which we hope will be found in future work.

The examples
From now on, F will denote a real quadratic field of narrow class number one. We let D be its fundamental discriminant. We will denote its ring of and K f be the order and the field generated by the Fourier coefficients, respectively. We will focus on forms such that [K f : Q] = 2, since we do not yet know how to write simple equations for general Hilbert modular varieties. We let D be the discriminant of K f and write e = √ D or (1 + √ D )/2. We denote the non-trivial element of Gal(F/Q) and Gal(K f /Q) by σ and τ respectively. The L-series of the conjectural surface A f attached to f is written as Our examples can be subdivided in the following cases, with the majority of examples coming from Case II. We will see that Case I is somewhat special: it is frequently possible to produce the associated abelian surface through analytic methods for classical modular forms.
In [6], Brumer-Kramer proposed the following conjecture as a genus 2 analogue of the Eichler-Shimura construction for classical newforms of weight 2 (with integer coefficients). The examples in Case II show that there is a strong connection between this conjecture and Conjecture 1. This implies that the form f is a base change from Q. Let g be a newform in S 2 (Γ 1 (D)) whose base change is f . Since the level of f is (1) . Let L g be the coefficient field of g. Then, L g is a quartic CM field which contains K f . The non-trivial element of Gal(L g /K f ), which we denote by (x →x, x ∈ L g ), extends to complex conjugation. Let B g be the abelian variety attached to the form g. Then B g is a fourfold such that End Q (B g ) ⊗ Q L g . Let w D be the Atkin-Lehner involution on S 2 (Γ 1 (D), χ D ) new . This induces an involution on B g , which we still denote by w D . Shimura [41, § 7.7] shows the following: (a) w D is defined over F , and w σ The abelian surface A f := (1 + w D )B g is defined over F , and is isogenous to its Galois conjugate given by Although Conjecture 1 is known in this case, it would still be desirable to have an explicit equation for the surface A f . We outline two methods to find it, the first of which is special to this case. , there exist newforms g 1 , g 2 ∈ S 2 (Γ 1 (D), χ D ) new such that g 1 , g 1 , g 2 and g 2 form a basis of the Hecke constituent of g and where a D (g) is the Hecke eigenvalue of g at D and λ D (g) = a D (g) √ D , the pseudo-eigenvalue of w D . The matrix of w D in the basis {g 1 ,ḡ 1 , g 2 ,ḡ 2 } is given by The following lemma is a simple adaptation of Cremona [8,Lemma 5.6.2].
, i = 1, 2, are bases for the ±-eigenspaces of W D , acting on the Hecke constituent of g, which give a decomposition of the space of 1-differential forms H 0 (B g ⊗ Q F, Ω 1 Bg⊗ Q F/F ) according to the action of Gal(F/Q).
Let H 1 (B g , Z) ± denote the ±-eigenspaces of w D . They are free Hecke submodules of H 1 (B g , Z) of rank 4 over Z, which are direct summands.
Lemma 4. Let Λ ± g be the period lattices obtained by integrating the forms in Then, there exist an abelian fourfold B g defined over Q, and an isogeny φ : B g → B g whose degree is a power of 2, such that B g (C) := C 4 /Λ g . Moreover, Proof. We first note that the complex tori C 2 /Λ ± g and C 4 /Λ g have canonical Riemann forms obtained by restriction of the intersection pairing ·, · on B g . Therefore, they are the complex points of some abelian varieties. Since h + 1 , h + 2 , h − 1 , h − 2 is a basis of the Hecke constituent of g, [41, Theorem 7.14 and Proposition 7.19] imply that there exist a fourfold B g defined over Q, and an isogeny φ : B g → B g , such that B g (C) = C 4 /Λ g .
Next, let x ∈ H 1 (B g , Z), then we have Hence the exponent of This implies that the degree of φ is a power of 2.
Since w D is defined over F and w σ D = −w D , the bases {h + 1 , h + 2 } and {h − 1 , h − 2 } are Gal(F/Q)-conjugate. Therefore C 2 /Λ + g and C 2 /Λ − g are the complex points of some abelian surfaces defined over F that are Galois conjugate. Let A f be the surface such that A f (C) = C 2 /Λ + g . Then, we see that B g = Res F/Q A f by construction.
In practice, we can replace B g by B g , and hence assume that . The above integration then gives the period lattice decomposition Provided that the intersection pairing restricted to H 1 (A f , Z) and H 1 (A σ f , Z) induces principal polarizations, we can compute the surfaces A f and A σ f as Jacobians of curves C f and C σ f (defined over F ). We illustrate this with the following example. The smallest discriminant for which we obtain a surface which satisfies Case I is D = 53. The abelian surface A f has real multiplication by (an order in) the field Q( √ 2). In fact, we will see that it has real multiplication by the full ring of integers.
A sympletic basis for H 1 (B g , Z) is given by the modular symbols In this case, we verify that the index of Z) is 4, and that the restriction of the intersection pairing to each direct summand H 1 (B g , Z) ± is of type (1,2). This means that A f and A σ f are not principally polarized with respect to the Riemann form given by the restriction of the intersection pairing from B g . Let G ± be the corresponding matrices for these pairings. We remedy this situation as follows. The element u = −e − 2 ∈ O f has norm 2, and acts on H 1 (B g , Z) ± as the Hecke operator T ± 7 . Letting G ± u = T ± 7 · G ± , we obtain principal polarizations on A f and A σ f by [24,Proposition 3.11].
By integrating the bases of differential forms respectively, we obtain the Riemann period matrices Ω A f and Ω A σ f , where where b = √ 53. By using Mestre's algorithm [34] which is implemented in Magma, we obtain a curve with above invariants. We reduce this curve using the algorithm in [3] implemented in Sage [37] to get the curve C f : y 2 = (−6w + 25)x 6 + (−60w + 246)x 5 + (−242w + 1017)x 4 + (−534w + 2160)x 3 + (−626w + 2688)x 2 + (−440w + 1724)x − 127w + 567, for which the Frobenius data of the corresponding Jacobian matches that of the Hilbert modular form, up to quadratic twist. Remark 1. We computed the curve C f by using the normalized period matrix Z. We could have instead applied the Jacobian nullwerte method [23,25] to the periods matrices Ω A f and Ω A σ f . This has the advantage of producing curves with small coefficients, needing no further reduction.  2 for a quick review of the results we need here). As a double-cover of P 2 r,s , it is given by z 2 = 2(16rs 2 + 32r 2 s − 40rs − s + 16r 3 + 24r 2 + 12r + 2).
The Igusa-Clebsch invariants (I 2 : I 4 : I 6 : I 10 ) ∈ P 2 (1:2:3:5) are given by where A 1 = 2rs 2 , A = −(9rs + 4r 2 + 4r + 1)/3, Recall that we expect to find a point of Y − (8) over F = Q( √ 53), corresponding to the principally polarized abelian surface A which should match the Hilbert modular form f . We first make a list of all F -rational points of height ≤ 200 on the Hilbert modular surface. Next, for each of these rational points, we try to construct the corresponding genus 2 curve C over F , whose Jacobian corresponds to the moduli point (r, s) we have chosen, and check whether the characteristic polynomial of Frobenius on its firstétale cohomology group matches up the polynomial Q p (T ) giving the corresponding Euler factor of surface A f attached to the Hilbert modular form. If a candidate point (r, s) passes this test for say the first 50 primes (ordered by norm) of F of good reduction for f and A = J(C), we can be reasonably convinced that it is the correct curve, and then try to prove that A is associated to f .
There are two subtleties in the search. First, since the Hilbert modular surface Y − (D ) is only a coarse moduli space, the point (r, s) is not enough to recover the curve up to F -isomorphism. The Igusa-Clebsch invariants are rational functions in r and s, and they are only enough to pin down C up to quadratic twist. Therefore, when we match the quartic L-factors L p (A, T ) and Q p (T ), we need to allow for rather than just the plus sign. Second, the Igusa-Clebsch invariants do not always allow us to define C over the base field F ; there is often a Brauer obstruction. Even when C is definable over F (which is the case we are interested in), it can be computationally expensive to do so. Therefore, it is convenient to speed up the process of testing compatibility with f by first reducing (I 2 , I 4 , I 6 , I 10 ) modulo p (assuming good reduction) and then producing a curve D p over F q from these reduced invariants, where q = Np. If C exists over F , then its reduction C p will be the same as D p up to quadratic twist. The advantage is that the Brauer obstruction vanishes over the finite field F p , making it very easy to check compatibility at p.
In this particular example, a search of Y − (8) for all points of height ≤ 200 using [15]  This leads to the same curve C f as above.
By further reducing the curve we obtained by either of Methods 1 or 2, we get the following.

Theorem 5.
Let C = C f : y 2 + Q(x)y = P (x) be the curve over F , where

Then
(a) The discriminant of this curve is ∆ C = − 7 . Thus C has everywhere good reduction. Proof. A direct calculation shows that ∆ C = − 7 . By construction, A has real multiplication by O f = Z[ √ 2], where 7 is split. Let λ be one of the primes above 7, and consider the λ-adic representation and its reductionρ modulo λ. We will show that ρ is modular by using [46,Theorem A]. For this, it suffices to show thatρ is reducible or, equivalently, that A has a 7-torsion point defined over F . By definition, we have So it is enough to find a degree zero divisor D defined over F such that 7D is principal. To this end, we consider the field L = F (α), where α is a root of the polynomial x 2 − wx + 3. Let σ ∈ Gal(L/F ) be the non-trivial involution. Then, the point P = (α, (−6w − 12)α + 2w + 18) ∈ C(L), and the divisor D := P + σ (P ) − 2∞ belongs to Pic 0 (C)(F ). An easy calculation shows that 7D ∼ (0). Hence, D corresponds to a point of order 7 in A(F ).
Since S 2 (1) has dimension 2 and is spanned by [f ], A must correspond to this Hilbert newform. Remark 4. The modularity of the abelian surface A = Jac(C) we found means that it is isogenous to the surface A f obtained from the Eichler-Shimura construction over Q. Since A f is a Q-surface, so is A. In fact, the proof of the reducibility ofρ A,λ implies that A and its Galois conjugate are related by a 7-isogeny.

4.2.
Case II. The following result explains the connection between Conjectures 1 and 2.
Proposition 6. Assume that Conjecture 2 is true. Let F be a real quadratic field. Let f be a Hilbert newform of weight (2,2) and level N over F , which satisfies the hypotheses of Case II. Then f satisfies Conjecture 1.
Proof. Since f is a non-base change, [26,Main Theorem] implies that there is a paramodular Siegel newform g of genus 2, level N D 2 and weight 2 attached to f , where N = N F/Q (N). Moreover, since Gal(F/Q) preserves {f, f τ }, we must have for all primes p ⊆ O F . Therefore, the Hecke eigenvalues of the form g are integers. So by Conjecture 2, there is an abelian surface B g defined over Q with End Q (B g ) = Z such that L(B g , s) = L(g, s). Let A f be the base change of B g to F . Then, by construction, we have s).
Hence, A f satisfies Conjecture 1.
Remark 5. By Proposition 6, if A f is an abelian surface attached to a Hilbert newform f satisfying Case II, then A f is the base change to F of some surface B defined over Q, which acquires extra endomorphisms. Therefore, we know that the Igusa-Clebsch invariants of A f are in Q, and we can use this fact in looking for A f .
(b) The surface J(C) is modular and corresponds to the form f listed in Table 3.

Remark 6.
A theorem of Stroeker [47] implies 1 that if E is an elliptic curve defined over a real quadratic field F having good reduction everywhere, then ∆ E / ∈ {−1, 1}. However, this fails for curves of genus 2, by the above example.
Proof. We show that ∆ C = −1 as before, which implies that C and J(C) both have everywhere good reduction. However, it is important to observe that we located the curve based on our heuristics which rely on Conjectures 1 and 2. Indeed, let S 2 (1) be the space of Hilbert cuspforms of level (1) and parallel weight 2 over F = Q( √ 193). Then S 2 (1) has dimension 9, and decomposes into two Hecke constituents of dimension 2 and 7 respectively. The form f in Table 3 is an eigenvector in the 2-dimensional constituent, and it is a non-base change whose Hecke constituent is Galois stable. So we can look for our surface A f with the help of Proposition 6.
To prove modularity, we note that 3 is inert in K f = Q( √ 17), and consider the 3-adic representation attached to A, By computing the orders of Frobenius for the first few primes, we see that the mod 3 representation is surjective, and absolutely irreducible. Hence ρ A,3 is also absolutely irreducible. Since 3 and 5 are unramified in the quadratic field F , the ramification indices ofρ A,3 at the primes of F above them are odd. Also, sinceρ A,3 is unramified at (5), the image of I (5) in SL 2 (F 9 ) is trivial, hence has odd order. By studying the Tate module of A × F F (ζ 3 ), we also see thatρ| G F (ζ 3 ) is also absolutely irreducible. Therefore,ρ A,3 is modular by [18,Theorem 3.2 and Proposition 3.4]. We then apply [21,22,Theorem 1.1 in Erratum] to conclude that ρ A,3 is modular. So, A is modular and corresponds to the Corollary 8. Let B be the Jacobian of the curve C /Q in the proof of Theorem 7. Then B is paramodular of level 193 2 .
Remark 7. In [6], the authors remarked that Conjecture 1.4 in their paper should be verifiable by current technology for paramodular abelian surfaces B over Q with End Q (B) Z. The majority of the surfaces we found fall in Case II (see section 5), and provide such evidence by Corollary 8.
In contrast to the curve in Theorem 7, we found a few curves whose Jacobians had everywhere good reduction while the curves themselves did not. We now discuss one such example, for the field F = Q( √ 929), with Hecke eigenvalues in Q( √ 13).   Table 4.
Proof. The curve C is a quadratic twist (by a 3-unit) of the curve C /Q given by We compute the reduction C of C at 3 by combining [32, Theorem 1 and Proposition 2], and Liu's algorithm implemented in Sage. This returns the type (V), [I * 0 − I * 0 − 0]. So the reduction A of the Jacobian A of C is a product of two elliptic curves whose j-invariants are j 1 = j 2 = 0 ([32, Theorem 1, (V)]). This implies that A has non-ordinary good reduction at 3. Therefore, A also has non-ordinary good reduction at (3). (Note that this is consistent with the fact that a (3) (f ) = 3.) Since 3 is the only prime dividing ∆ C , we see that A has everywhere good reduction.
To prove modularity, we recall that by construction A has real multiplica- 13 2 ], where 3 splits. We choose a prime λ above 3, and consider the λ-adic representation and its reductionρ A,λ modulo λ. By computing the first few Frobenii, we see thatρ A,λ is surjective, hence irreducible. Since GL 2 (F 3 ) is solvable,ρ is modular by Langlands-Tunnell. By looking at the Tate module of A× F F (ζ 3 ), we also see that ρ A,λ is not induced from F (ζ 3 ). So, we conclude that ρ A,λ is modular by [21,22, Theorem 1.1 in Erratum]. Remark 8. The example in Theorem 9 and other similar ones in Table 7 underscore the difficulty in producing effective algorithms for principally polarized abelian surfaces with good reduction outside a (finite) prescribed set of primes S of O F . Indeed, let A be such a surface so that A = Jac(C), where C is a curve defined over F with good reduction outside a finite set of primes T ⊇ S. Then, the set T \ S is non-empty in general, depends a priori on A, and is hard to predict. When A has real multiplication by some quadratic field K and is attached to a modular form f , T \ S is contained in the set of non-ordinary primes for f , which may still be infinite.
Similar proofs apply for the other Hilbert modular forms in Case II for which we were able to find matching principally polarized abelian surfaces. However, there are five examples (listed in Table 5) for which we were unable as yet to find matching abelian surfaces. In each case, the Fourier coefficients of the form indicate that the missing surface would have real multiplication by the full ring of integers O D . So, assuming the Eichler-Shimura conjecture holds, our difficulties in matching those forms could be due to one of the following reasons: (a) Our height bound for the rational point search on the corresponding Hilbert modular surfaces is too small. We searched for parameters r, s ∈ Q of height up to 1000. (b) The corresponding abelian surface is not principally polarized. Note that the criteria given in [24,Proposition 3.11] to convert an arbitrary polarization to a principal polarization fail for each of the missing discriminants D . For (D, D ) = (677, 85), the field Q( √ D ) has class number 2, whereas for the other examples, there is no unit of negative norm. There is also the possibility, since the models in [17] are birational to Y − (D ) (rather than isomorphic), that we might have missed some curves or points in our search. However, this is unlikely to be the case, as the extra points should correspond to abelian surfaces with extra endomorphisms.

4.3.
Case III. This is by far the trickiest case, since the Igusa-Clebsch invariants (and therefore r, s) are not in Q. This leads to a much slower search for F -points on Y − (D ), compared to searching for Q-points. We searched for points of height up to 400 using the enumeration of points of small height developed in [15] (implemented in Sage), but were unable to find either of the two examples predicted by the Eichler-Shimura conjecture, corresponding to the Hilbert modular forms of level 1 and weight 2 over Q( √ 353) and Q( √ 421), both with Fourier coefficients in Q( √ 5). In addition to the reduced search height bound, another complicating factor is the fundamental unit of F , which might be quite large. In Case II, the discriminant of the genus 2 curve differed from I 10 (r, s) by only a few small (rational) primes. However, in Case III, one has to take into account the fact that a power of the fundamental unit might also appear in the discriminant. On the other hand, principal polarizability is not an obstruction, as Q( √ 5) has a fundamental unit of negative norm.
We hope to address the missing examples using different techniques in future work.

The data
Below we list genus 2 curves y 2 + Q(x)y = P (x) matching the data. We always set b = √ D and w = (b + 1)/2. We suppress Q(x) when it is 0.