EMPIRICAL EVIDENCE FOR THE BIRCH AND SWINNERTON-DYER CONJECTURES FOR MODULAR JACOBIANS OF GENUS 2 CURVES

E. VICTOR FLYNN, FRANCK LEPRÉVOST, EDWARD F. SCHAEFER, MICHAEL STOLL, AND JOSEPH L. WETHERELL

Abstract. This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of those six quantities is the size of the Tate-Shafarevich group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer, which is a power of 2. In addition, this power of 2 is as predicted by the 2-torsion of the Tate-Shafarevich group, which we could compute.

1. Introduction

In this article, we provide empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. Such evidence has already been provided for elliptic curves (see [BSD, Ca, Cr]). We will always consider our base field to be the rational numbers.

There are two Birch and Swinnerton-Dyer conjectures for Jacobians of genus 2 curves. These were formulated by Tate in [Ta]. The first is that the order of the zero of the L-series of the Jacobian at s = 1 (the analytic rank) is equal to the Mordell-Weil rank of the Jacobian. Since our Jacobians are modular, their L-series have analytic continuations to s = 1 and hence the analytic rank is defined. The second conjecture is that

(1)  lim_{s→1}(s − 1)^{−r}L(J, s) = Ω · Reg · ∏_{p} c_{p} · #� (J, Q) · (#J(Q)_{tors})^{−2}.

In the above equation, L(J, s) is the L-series of the Jacobian J. We use r to denote the rank. We say that a differential form on J is integral if it is the pullback to J of a global holomorphic form on the Néron model of J over Z. With this notation, Ω denotes the integral over J(R) of the generator of the integral 2-forms that makes Ω positive. Reg is the regulator of J(Q). For primes p, we use c_{p} to denote the size of J(Q_{p})/J^{0}(Q_{p}), where J^{0}

Date: April 22, 1999. 1991 Mathematics Subject Classification. Primary 11G40; Secondary 11G10, 11G30, 14H25, 14H40,14H45. Key words and phrases. Birch and Swinnerton-Dyer conjecture, genus 2, Jacobian, modular abelian

variety.

The first author thanks the Nuffield Foundation (Grant SCI/180/96/71/G) for financial support. The second author did some of the research at the Max-Planck Institut für Mathematik and the Technische Universität Berlin. The third author thanks the National Security Agency (Grant MDA904-95-H-1051). The fourth author did some of the research at the Max-Planck-Institut für Mathematik. The fifth author thanks the National Science Foundation (Grant DMS-9705959). The authors had useful conversations with John Cremona, Qing Liu, William Stein and Peter Swinnerton-Dyer and are grateful to Xiangdong Wang and Michael Müller for making data available to them.

will be defined in section 3.5. We let � (J, Q) be the Tate-Shafarevich group of J over Q; it is assumed to be finite. We use J(Q)_{tors} to denote the torsion subgroup of J(Q).

The determination of both kinds of rank and the five computable terms of the above equation required several different algorithms. Our policy in this article is the following. If one of these algorithms is already well described in the literature, then we simply cite it. The rest, we describe here. The modular Jacobians of genus 2 curves that we consider here come from two sources to be described in section 2.

The first Birch and Swinnerton-Dyer conjecture has been proven for modular abelian varieties associated to newforms whose L-series have analytic rank 0 (see [KL]) or analytic rank 1. The Tate-Shafarevich group has also been shown to be finite in the cases of analytic rank 0 (see [KL]) and analytic rank 1. For both results, the rank 1 case is apparently wellknown among the experts, but not written down anywhere. In our case, the L-series of the Jacobian is the product of the L-series of a newform and its conjugate. Therefore, when the analytic rank of the L-series of the newform and its conjugate is 0 or 1, then the analytic rank of the L-series of the Jacobian is 0 or 2, respectively. All of our Mordell-Weil ranks were 0 or 2 and agreed with the analytic ranks. Thus we can not provide any new evidence for the first conjecture.

The second Birch and Swinnerton-Dyer conjecture has not been proven for the cases we consider. In order to verify equation (1), we would need to compute #� (J, Q). Though we were not able to do that, we were able to determine the 2-torsion of each � (J, Q). We computed the five terms other than #� (J, Q) and solved for #� (J, Q). In each case, the value is an integer to within the accuracy of our calculations. The size of each � (J, Q) is a power of 2. In addition, this power of 2 is compatible with the independently computed 2-torsion of � (J, Q).

2. The Curves

We will denote each curve by C_{N} where N is the level from which it arose. When there is more than one curve arising from the level N, then we denote them C_{N,A}, C_{N,B}, etc. The curves we use come from two sources. Hasegawa [Hs] has provided all genus 2 quotients of modular curves of the form X_{0}(N) by a subgroup of the Atkin-Lehner involutions. There are 142 such curves. Their Jacobians correspond to 2-dimensional subspaces of S_{2}(N), the space of cusp forms of weight 2 for Γ_{0}(N). We are particularly interested in those where this subspace is spanned by a pair of quadratic conjugate newforms. There are 21 of these with level N < 200. These are the curves listed in Table 1 with an H in the last column. For these curves we will provide evidence for the second conjecture. There are 7 more such curves with N > 200. We can classify the other 2-dimensional subspaces into four other kinds. There are 2-dimensional subspaces of oldforms that are irreducible under the action of the Hecke algebra. There are also 2-dimensional subspaces that are reducible under the action of the Hecke algebra and are spanned by two oldforms, two newforms or one of each. The Jacobians corresponding to the latter three kinds are always isogenous, over Q, to the product of two elliptic curves. Given the small levels, these are elliptic curves for which Cremona [Cr] has already provided evidence for the Birch and Swinnerton-Dyer conjectures. In Table 5, we describe the kind of cusp forms spanning the 2-dimensional subspace and the signs of their functional equations from the level at which they are newforms. The analytic and Mordell-Weil ranks were always the minimal possible given those signs.

The second set of curves was created by Wang [Wa] and is further discussed in [FM]. This set consists of 28 curves that were constructed by considering the curves X_{0}(N) with N ≤ 200. For some of these modular curves, the spaces S_{2}(N) contain subspaces spanned by a pair of quadratic conjugate newforms. These were integrated to produce an abelian variety. The period matrix for the abelian variety was created using certain intersection numbers. When those intersection numbers are the same, then the abelian variety is principally polarized. The converse is not always true. Using Igusa invariants, real approximations were found for the equations of the curves associated to the principally polarized abelian varieties. The real coefficients were all close approximations to rational numbers of small heights. Wang’s equations for the curves used these rational numbers. In 17 of these cases, the 2-dimensional subspace spanned by the newforms is the same as that giving one of Hasegawa’s curves. In all of those cases, the curve given by Wang’s equation is isomorphic, over Q, to that given by Hasegawa. Those are the curves listed in Table 1 with HW in the last column.

For the remaining 11 curves (listed in Table 1 with a W in the last column), we want to convince ourselves that the Jacobians of these curves are, indeed, the appropriate factors of the Jacobians of the corresponding modular curves. In what follows, we will use the notation of [Cr] and recommend it as a reference on the general results that we assume here and in sections 3.1 and 6. Fix a level N and let f(z) ∈ S_{2}(N). Then f has a Fourier expansion

f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}.

For a newform f, we have a_{1} ≠ 0; so we can normalize it by setting a_{1} = 1. In our cases, the a_{n}’s are defined over a real quadratic field. For each prime p not dividing N, the corresponding Euler factor of the L-series L(f, s) is 1 − a_{p}p^{−s} + p^{1−2s}. Let N(a_{p}) and T r(a_{p}) denote the norm and trace of a_{p}. The product of this Euler factor and its conjugate is 1 − T r(a_{p})p^{−s} + (N(a_{p}) + 2p)p^{−2s} − pT r(a_{p})p^{−3s} + p^{2}p^{−4s}. Therefore, the characteristic polynomial of the p-Frobenius on the corresponding abelian variety over F_{p} is x^{4}−T r(a_{p})x^{3}+ (N(a_{p}) + 2p)x^{2} − pT r(a_{p})x + p^{2}. Let C be a curve, over Q, whose Jacobian, over Q, comes from the space spanned by f and its conjugate. Then we know that p+1−#C(F_{p}) = T r(a_{p}) and \frac{1}{2}(#C(F_{p})^{2} + #C(F_{p^{2}})) − (p + 1)#C(F_{p}) − p = N(a_{p}) (see [MS, Lemma 3]). For the odd primes less than 200, not dividing N, we computed #C(F_{p}) and #C(F_{p^{2}}) for each curve given by one of Wang’s equations. From these we could compute the characteristic polynomials of Frobenius and see if they agreed with those predicted by the a_{p}’s of the newforms.

Of the 11 curves, the characteristic polynomials agreed for only four. In the remaining seven cases, we found quadratic twists of four of the curves and higher degree twists of the other three curves, whose characteristic polynomials did agree with those predicted by the newforms for all odd primes less than 200 not dividing N. It is those twists that appear in Table 1.

We can provide further evidence that these equations are correct. For each curve, given in Table 1, it is easy to determine the primes of singular reduction. In section 3.5, we will provide techniques for determining which of those primes divides the conductor of its Jacobian. In each case, the primes dividing the conductor of the Jacobian of the curve are exactly the primes dividing the level N; this is necessary. With the exception of curve C_{188}, all the curves come from odd levels. We used Liu’s genus2reduction program

(ftp://megrez.math.u-bordeaux.fr/pub/liu) to compute the conductor of the curve. In

each case (other than curve C_{188}), the conductor is the square of the level; this is also necessary. For curve C_{188}, the odd part of the conductor of the curve is the square of the odd part of the level.

The Jacobians are absolutely simple and the canonically polarized Jacobians have automorphism groups of size two with the exception of curves C_{63}, C_{117,A} and C_{189}. We showed that these Jacobians are absolutely simple using an argument like those in [Le, St1]. The automorphism group of the canonically polarized Jacobian of a hyperelliptic curve is isomorphic to the automorphism group of the curve (see [Mi2, Thm. 12.1]). Each automorphism of a hyperelliptic curve is induced by a linear fractional transformation on x-coordinates (see [CF, p. 1]). Each automorphism also permutes the six Weierstrass points. Once we believed we had found all of the automorphisms, we were able to show that there were no more by considering all linear fractional transformations sending three fixed Weierstrass points to any three Weierstrass points. In each case, we worked with sufficient accuracy to show that other linear fractional transformations did not permute the Weierstrass points.

The Jacobians of curves C_{63}, C_{117,A} and C_{189} are each isogenous to the product of two elliptic curves (though not over Q where they are simple) and these genus 2 curves have automorphism groups of size 12. In the following table, we list the curve at the left. On the right, we give one of the elliptic curves which is a factor of its Jacobian. The second factor is the conjugate where ζ_{3} is a primitive third root of unity.

\begin{matrix}C_{63} : & y^{2} = x(x^{2} + (9 − 12ζ_{3})x − 48ζ_{3}) \\ C_{117,A} : & y^{2} = x(x^{2} − (12 + 27ζ_{3})x − (48 + 48ζ_{3})) \\ C_{189} : & y^{2} = x^{3} + (66 − 3ζ_{3})x^{2} + (342 + 81ζ_{3})x + 105 + 21ζ_{3} \end{matrix}

The a_{p}’s and other information concerning Wang’s curves are currently kept in a database at the Institut für experimentelle Mathematik in Essen, Germany. Most recently, this database was under the care of Michael Müller.

3. Algorithms

In this section, we describe the algorithms used to determine the data necessary to provide evidence for the conjectures. The computation of the analytic rank of L(J, s) and of lim_{s→1} L(J, s)/(s − 1)^{r} used the newforms. All other algorithms used the given equations for the curves and so their outputs are only conjectural for the 11 curves for which we cannot prove that we have the right equation.

3.1. Analytic rank of L(J, s) and leading coefficient at s = 1. Fix a Jacobian J corresponding to the 2-dimensional subspace of S_{2}(N) spanned by quadratic conjugate, normalized newforms f and \overline{f}. Let W_{N} be the Fricke involution. The newforms f and \overline{f} have the same eigenvalue ϵ_{N} with respect to W_{N}, namely 1 or −1. In the notation of section 2, let

L(f, s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}

be the L-series of f; then L(\overline{f}, s) is its conjugate. The coefficients of L(\overline{f}, s) are the conjugates of the coefficients of L(f, s). The order of L(f, s) at s = 1 is even when ϵ_{N} = −1 and odd when ϵ_{N} = 1. We have L(J, s) = L(f, s)L(\overline{f}, s). Thus the analytic rank of J is 0 modulo 4 when ϵ_{N} = −1 and 2 modulo 4 when ϵ_{N} = 1. We found that the ranks were all 0 or 2. To prove that the analytic rank of J is 0, we need to show L(f, 1) ≠ 0 and L(\overline{f}, 1) ≠ 0. In the case that ϵ_{N} = 1, to prove that the analytic rank is 2, we need to show that L′(f, 1) ≠ 0 and L′(\overline{f}, 1) ≠ 0. When ϵ_{N} = −1, we can evaluate L(f, 1) as in [Cr, §2.11]. When ϵ_{N} = 1, we can evaluate L′(f, 1) as in [Cr, §2.13]. Each appropriate L(f, 1) or L′(f, 1) was at least 0.1 and the errors in our approximations were all less than 10^{−67}. In this way we determined the analytic ranks which we denote r.

To compute the leading coefficient of L(J, s) at s = 1, we note that lim_{s→1} L(J, s)/(s−1)^{r} = L^{(r)}(J, 1)/r!. In the r = 0 case, we simply have L(J, 1) = L(f, 1)L(\overline{f}, 1). In the r = 2 case, we have L′′(J, s) = L′′(f, s)L(\overline{f}, s)+2L′(f, s)L′(\overline{f}, s)+L(f, s)L′′(\overline{f}, s). Evaluating both sides at s = 1 we get \frac{1}{2}L′′(J, 1) = L′(f, 1)L′(\overline{f}, 1).

3.2. Torsion Subgroup. Computing the torsion subgroup of J(Q) is straightforward. We used the technique described in [CF, pp. 78-82]. This technique is not always effective, however. For an algorithm working in all cases see [St3].

3.3. Mordell-Weil rank and � (J, Q)[2]. The group J(Q) is a finitely generated abelian group and so is isomorphic to Z^{r} ⊕ J(Q)_{tors} for some r called the Mordell-Weil rank. It is an abuse of notation to use r to denote both the Mordell-Weil and analytic ranks; we do so since they were always the same. We used the algorithm described in [FPS] to compute Sel_{fake}^{2}(J, Q) (notation from [PSc]), which is a quotient of the 2-Selmer group, Sel^{2}(J, Q). More details on this algorithm can be found in [St2]. Theorem 13.2 of [PSc] explains how to get Sel^{2}(J, Q) from Sel_{fake}^{2}(J, Q). Let M[2] denote the 2-torsion of an abelian group M and let dimV denote the dimension of an F_{2} vector space V. We have dim Sel^{2}(J, Q) = r + dimJ(Q)[2] + dim � (J, Q)[2].

Recall that for the 32 curves under consideration, the analytic ranks were 0 or 2. In the analytic rank 2 cases, we can always find two independent elements of J(Q) of infinite order. So we know that r ≥ 2. In each of these cases, dim Sel^{2}(J, Q) − dimJ(Q)[2] = 2. Therefore we know both that the Mordell-Weil rank is exactly 2 for the curves given in Table 1, and that dim � (J, Q)[2] = 0.

For the analytic rank 0 cases, we also computed Sel^{2}(J, Q). We know from [KL] that the Mordell-Weil rank of the Jacobians must be 0 also. In these cases we have dim � (J, Q)[2] = dim Sel^{2}(J, Q) − dimJ(Q)[2]. In all but four of these cases, dim � (J, Q)[2] = 0 and so we were able to prove that the Mordell-Weil rank is 0 for the curve given in Table 1. Those four cases are among the 11 where we can not prove that the equation for the curve, given in Table 1, is correct. In section 5, we will prove that the Mordell-Weil rank is 0 for the Jacobians of two of the remaining four curves.

3.4. Regulator. When the Mordell-Weil rank is 0, then the regulator is 1. When the Mordell-Weil rank is positive, then to compute the regulator, we first need to find generators for J(Q)/J(Q)_{tors}. The regulator is the determinant of the canonical height pairing matrix on this set of generators. An algorithm for computing the generators and canonical heights is given in [FS]. In that article, the algorithm for computing height constants at the infinite prime is not clearly explained and there are some errors in the examples. A clear algorithm for computing infinite height constants is given in [St3].

3.5. Tamagawa Numbers. Let O be the integer ring in K which will be Q_{p} or Q_{p}^{unr} (the maximal unramified extension of Q_{p}). Let J be the Néron model of J over O. Define J^{0} to be the open subgroup scheme of J whose generic fiber is isomorphic to J over K and whose special fiber is the identity component of the closed fiber of J. The group J^{0}(O) is isomorphic to a subgroup of J(K) which we denote J^{0}(K). The group J(Q_{p}^{unr})/J^{0}(Q_{p}^{unr}) is the component group of J over O_{Q_{p}^{unr}} and #J(Q_{p}^{unr})/J^{0}(Q_{p}^{unr}) is usually called the Tamagawa number. We are interested in computing c_{p} = #J(Q_{p})/J^{0}(Q_{p}). Since Néron models are stable under unramified base extension, the Gal(Q_{p}^{unr}/Q_{p})-invariant subgroup of J^{0}(Q_{p}^{unr}) is J^{0}(Q_{p}). Since H^{1}(Gal(Q_{p}^{unr}/Q_{p}), J^{0}(Q_{p}^{unr})) is trivial (see [Mi1, p. 58]) we see that the Gal(Q_{p}^{unr}/Q_{p})-invariant subgroup of J(Q_{p}^{unr})/J^{0}(Q_{p}^{unr}) is J(Q_{p})/J^{0}(Q_{p}).

There exist several discussions in the literature on constructing the group J(Q_{p}^{unr})/J^{0}(Q_{p}^{unr}) starting with an integral model of the underlying curve. For our purposes, we especially recommend Silverman’s book [Si], chapter IV, sections 4 and 7. There one can find justifications for what we will do. While constructing such groups, we ran into a number of difficulties that we did not find described anywhere. For that reason, we will present examples of such difficulties that arose in our examples, as well as our methods of resolution. We do not claim that we will describe all situations that could arise.

In order to compute c_{p}, we first find a minimal, proper, regular model C for C over O_{Q_{p}^{unr}}. We start with the models in Table 1. Technically, we consider the curves to be the two affine pieces y^{2} + g(x)y = f(x) and v^{2} + u^{3}g(1/u)v = u^{6}f(1/u), glued together by ux = 1, v = u^{3}y. We blow them up at all points that are not regular until we have a regular model. (A point is regular if the cotangent space there has two generators). These curves are all proper, and this is not affected by blowing up. When the special fiber does not contain an exceptional component (a component of genus 0 and self-intersection number −1), then the model is minimal. When the special fiber does contain an exceptional component, then we must blow it down. Example 1. Curve C_{65,B} over Z_{2}.

An equation for curve C_{65,B} is y^{2} = −x^{6} + 10x^{5} − 32x^{4} + 20x^{3} + 40x^{2} + 6x − 1. The Jacobian of this curve is a quotient of the Jacobian of X_{0}(65). Though 65 is odd, this curve has singular reduction at 2. Since the equation for this curve is really only conjectural, it will be nice to verify that 2 does not divide the conductor of its Jacobian, i.e. that the Jacobian has good reduction at 2. In addition, we will need a minimal, regular model for this curve in order to find Ω.

Note that −x^{6} +10x^{5} −32x^{4} +20x^{3} +40x^{2} +6x−1 has a factor of x^{2} −3x−1. The special fiber of the arithmetic surface y^{2} = −x^{6} + 5px^{5} − p^{5}x^{4} + 5p^{2}x^{3} + 5p^{3}x^{2} + 3px − 1, p = 2 over Z_{2} is a genus 0 curve of multiplicity 2. It is not regular at the two points (x − α, y, p, 2), where α is a root of x^{2} − 3x − 1, and at infinity.

When we try to blow up the point at infinity, we end up with a new component which is nowhere regular. When we blow it up, we get a component which is nowhere regular with higher multiplicity. It appeared that this would go on forever. What we found that worked was to use the following transformation.

(x_{1}, y_{1}) = (\frac{2(x − α + 1)}{x − α}, \frac{8y}{(x − α)^{3}} ) or (x, y) = (\frac{αx_{1} − 2α + 2}{x_{1} − 2}, \frac{y_{1}}{(x_{1} − 2)^{3}} ) .

The transformation x_{1} = 2(x − α + 1)/(x − α) has three effects. The denominator sends a Weierstrass point (which is not regular at 2) to infinity and thus produces a quintic equation. The factor (x − α + 1) in the numerator has the effect, on the reduction, of moving the other non-regular affine point to the origin. Since the limit as x approaches infinity of the expression is 2, the problematic point at infinity is also moved to the origin in the reduction.

The curve now has the form

y_{1}^{2} = (−48α − 32)x_{1}^{5} + (436α + 312)x_{1}^{4} + (−1792α − 736)x_{1}^{3}
+ (4208α − 1264)x_{1}^{2} + (−5632α + 6784)x_{1} + (3328α − 6656).

We would like to divide through by as high a power of 2 as possible. Note 2^{4} exactly divides the coefficient of x_{1}^{5} and divides all coefficients except for the coefficient of x_{1}^{4}, which 2^{2} divides exactly. When we make the substitution y_{1} = 4y_{2} + 2(α − 1)x_{1}^{2}, we get the equation

y_{2}^{2} + (α − 1)x_{1}^{2}y_{2} = (−3α − 2)x_{1}^{5} + (27α + 19)x_{1}^{4} + (−112α − 46)x_{1}^{3}
+ (263α − 79)x_{1}^{2} + (−352α + 424)x_{1} + 208α − 416

for our curve.

The equation above has the form y_{2}^{2} +h(x_{1})y_{2} = f(x_{1}). We get a complete curve by gluing together that affine model with the affine model v^{2} + u^{3}h(1/u)v = u^{6}f(1/u) using ux_{1} = 1 and y_{2}u^{3} = v. The point outside of the first affine model, namely (u, v) = (0, 0), is regular at 2.

For the first affine model, we describe the curve by

g_{1}(x_{1}, y_{2}, p) = (−3α − 2)x_{1}^{5} + (27α + 19)x_{1}^{4} + (−56α − 23)px_{1}^{3} + (263α − 79)x_{1}^{2}
+ (−44α + 53)p^{3}x_{1} + (13α − 26)p^{4} − y_{2}^{2} − (α − 1)x_{1}^{2}y_{2}
= 0,
p = 2.

The special fiber of this model of the curve consists of a single component which we denote A. It is regular at every point except (x_{1}, y_{2}, p, 2). We blow up at that point by introducing projective coordinates x_{3}, y_{3}, p_{3} with x_{1}y_{3} = x_{3}y_{2}, x_{1}p_{3} = x_{3}p, y_{2}p_{3} = py_{3}. We look in all three affine covers and check for regularity.

x_{3} = 1: y_{2} = x_{1}y_{3}, p = x_{1}p_{3}. We get g_{2}(x_{1}, y_{3}, p_{3}) = x_{1}^{−2}g_{1}(x_{1}, x_{1}y_{3}, x_{1}p_{3}) = 0 and x_{1}p_{3} = 2. In the reduction we have x_{1} = 0: (y_{3} + α)^{2} = 0. This is a new component which we denote B. It has genus 0 and multiplicity 2. We check regularity along B at (x_{1}, y_{3} + α, p_{3} − t, 2), with t in the integer ring of Q_{2}^{unr}, and find that B is regular at every point. p_{3} = 0: y_{3}^{2} + (α + 1)x_{1}y_{3} = αx_{1}^{3} + (α + 1)x_{1}^{2} + α + 1. Using the gluing maps we see that this is A. It has genus 1 and multiplicity 1 and is non-singular. Therefore it is regular at every point. The point where A and B intersect transversally is also regular.

y_{3} = 1: We get no new information from this affine cover.

p_{3} = 1: x_{1} = x_{3}p, y_{2} = y_{3}p. We get g_{3}(x_{3}, y_{3}, p) = p^{−2}g_{1}(x_{3}p, y_{3}p, p) = 0, and p = 2. In the reduction we have p = 0: (y_{2} + αx_{3})^{2} = 0. This is B and we find the point (x_{3}, y_{3}, p, 2) is not regular.

The current special fiber is in Figure 1 and is labelled Fiber 1. It is not regular at exactly one point.

We blow up the affine cover p_{3} = 1 at (x_{3}, y_{3}, p, 2) by introducing projective coordinates x_{4}, y_{4}, p_{4} with x_{3}y_{4} = x_{4}y_{3}, x_{3}p_{4} = x_{4}p, y_{3}p_{4} = y_{4}p. We look in all three affine covers and check for regularity.

x_{4} = 1: y_{3} = x_{3}y_{4}, p = x_{3}p_{4}. We get g_{4}(x_{3}, y_{4}, p_{4}) = x_{3}^{−2}g_{3}(x_{3}, x_{3}y_{4}, x_{3}p_{4}) = 0 and x_{3}p_{4} = 2. In the reduction we have Figure 1. Special fibers of curve C_{65,B} over Z_{2}; points not regular are thick

[figure: Fiber 1; A; B; 2; ^{Fiber 2 A D B2 2}; ^{A D B Fiber 3 2}; A; D; Fiber 4]

x_{3} = 0: (y_{4} + (α + 1)p_{4} + α)^{2} = 0. This is a new component we denote D. It has genus 0, multiplicity 2 and is nowhere regular. p_{4} = 0: (y_{4} + α)^{2} = 0. This is B. It crosses D transversally at (x_{3}, y_{4} − α, p_{4}, 2).

y_{4} = 1 and p_{4} = 1: We get no new information from these affine covers.

The current special fiber is in Figure 1 and is labelled Fiber 2. It is regular everywhere except for D where it is nowhere regular. The component D does not lie entirely in any one affine cover so we will blow up the affine covers x_{4} = 1 and p_{4} = 1 along D.

To blow up x_{4} = 1 along D we first make the substitution y_{5} = y_{4} + (α + 1)p_{4} + α and then replace each factor of 2 in a coefficient by x_{3}p_{4}. We have g_{5}(x_{3}, y_{5}, p_{4}) = 0 and x_{3}p_{4} = 2 and we want to blow up along the line (x_{3}, y_{5}, 2). We have g_{5}(x_{3}, y_{5}, p_{4}) = h_{1}(y_{5}, p_{4})x_{3}^{2} + h_{2}(y_{5}, p_{4})x_{3}y_{5} − y_{5}^{2}, for some integral polynomials h_{1}, h_{2}. This tells us that the valuation of the y_{5}-coordinate of a point is greater than that of the x_{3}-coordinate, so it is safe to make the replacement y_{5} = y_{6}x_{3}.

y_{5} = y_{6}x_{3}: We get g_{6}(x_{3}, y_{6}, p_{4}) = x_{3}^{−2}g_{5}(x_{3}, x_{3}y_{6}, p_{4}) = 0 and x_{3}p_{4} = 2. In the reduction we have x_{3} = 0: y_{6}^{2} + ((α + 1)p_{4}^{2} + αp_{4})y_{6} = p_{4}^{4} + p_{4}. This is D. It is now a non-singular genus 1 curve. p_{4} = 0: y_{6}^{2} = 0. This is B. It meets D transversally at (x_{3}, y_{6}, p_{4}, 2), where it is regular.

When we blow up p_{4} = 1 along D we get essentially the same thing and all points are again regular. The current special fiber is in Figure 1 and is labelled Fiber 3.

We now have a regular, proper model of C over Z_{2}. We check minimality by determining if there is an exceptional component, i.e. a curve of genus 0 and self-intersection number −1. The only component of the special fiber with genus 0 is B. Let C′_{2} denote the current special fiber. The intersection pairing of the special fiber with any of the components is 0. We note that

C′_{2} · B = 0 = (A + 2B + D) · B = A · B + 2B · B + D · B.

Since B meets both A and D transversally, both of those intersection numbers are 1. Therefore the self-intersection number of B is −1 and so B is exceptional and our model is not minimal. When we blow down, B disappears and we get A and D crossing transversally (see [Ha, §I.4]) and any element of Gal(\overline{Q_{2}}/Q_{2}) that does not fix the quadratic unramified extension of Q_{2} will switch A and D. The special fiber is in Figure 1 and is labelled Fiber 4. We currently have C (once all of the affines have been glued together), the minimal, proper, regular model for C over Z_{2}.

Let C_{2} denote the special fiber of C. The group J(Q_{2}^{unr})/J^{0}(Q_{2}^{unr}) is isomorphic to a quotient of the degree 0 part of the free group on the irreducible components of C_{2}. In our case, the degree 0 part of the free group has the form

L = {αA + δD | 1α + 1δ = 0}.

The numbers in front of α and δ are the multiplicities of A and D in the special fiber, which are both 1 in this case. The kernel of the map from the free group to J(Q_{2}^{unr})/J^{0}(Q_{2}^{unr}) is generated by divisors of the form

∑_{D∈C_{2}}(D′ · D)D

for a fixed irreducible component D′ of C_{2} and where the D range over the irreducible components of C_{2} (see [BL]). As we have already seen, we can deduce D′ · D′ by noting that the divisor must be contained in the group L.

In our case, those divisors are

\begin{matrix}D′ = A : & −A + D \\ D′ = D : & A − D. \end{matrix}

We have that the group J(Q_{2}^{unr})/J^{0}(Q_{2}^{unr}) is isomorphic to the group L modulo the subgroup generated by the divisors listed above. This is a trivial group. Therefore c_{2} = #J(Q_{2})/J^{0}(Q_{2}) = 1.

In fact, we see that the Jacobian of the reduction of the minimal model is the product of the elliptic curves A and D. So the Jacobian of the curve given by our equation has good reduction at 2. Since the conductor of the Jacobian is odd, this gives further evidence that the equation given in Table 1 is correct. Example 2. Curve C_{63} over Z_{3}.

The Tamagawa number is often found using the intersection matrix and sub-determinants. This is not entirely satisfactory for cases where the special fiber has several components and a non-trivial Galois action. Here is an example of how to resolve this (see also [BL]).

When we blow up curve C_{63} over Z_{3}, and blow down one exceptional component we get the special fiber shown in Figure 2. Elements of Gal(\overline{Q_{3}}/Q_{3}) that do not fix the quadratic unramified extension of Q_{3} switch G and H. The other components are defined over Q_{3}. All components have genus 0. The group J(Q_{3}^{unr})/J^{0}(Q_{3}^{unr}) is isomorphic to a quotient of

L = {αA + βB + δD + ϵE + ϕF + γG + ηH | α + β + 2δ + 4ϵ + 2ϕ + 2γ + 2η = 0}.

The kernel is generated by the following divisors. D′ = A : −2A + D D′ = B : −2B + D D′ = D : A + B − 3D + E D′ = E : D − 2E + F + G + H D′ = F : E − 2F D′ = G : E − 2G D′ = H : E − 2H

We have that the group J(Q_{3}^{unr})/J^{0}(Q_{3}^{unr}) is isomorphic to the group L modulo the subgroup generated by the divisors listed above. When we project away from A, we find that this group is isomorphic to

⟨B, D, E, F, G, H | D = 0, D = 2B, 3D = B + E,
2E = D + F + G + H, E = 2F = 2G = 2H⟩.

Figure 2. Special fiber of curve C_{63} over Z_{3}

F G H  D  E A B 2 2 2  4 2

At this point, it is straightforward to simplify the representation by elimination. Note that we projected away from A, which is Galois-invariant. It is best to continue eliminating Galoisinvariant elements first. We find that this group is isomorphic to ⟨G, H | 2G = 2H = 0⟩ and elements of Gal(\overline{Q_{3}}/Q_{3}) that do not fix the quadratic unramified extension of Q_{3}, switch G and H. Therefore #J(Q_{3}^{unr})/J^{0}(Q_{3}^{unr}) = 4 and c_{3} = #J(Q_{3})/J^{0}(Q_{3}) = 2.

3.6. Computing Ω. Recall that by an integral differential (or integral form) on J we mean the pullback to J of a global relative differential form on the Néron model of J over Z. The set of integral n-forms on J is a lattice in the (vector space of) global holomorphic n-forms on J. Since J has dimension 2, the integral 2-forms are a free Z-module of rank 1. The quantity Ω is the integral, over the real points of J, of a generator of the integral 2-forms. (We choose the generator which leads to a positive integral.)

Because of the identifications of the homologies and cohomologies of C and J (see [Mi2]), and because of the close connection between the Néron model of J and the Picard group of a regular proper model of C, we can compute Ω using C in the following way. Choose a regular proper model for C over Z and find generators of the global relative differentials for this model; there will be two. We will call this a basis for the integral differentials on C. Find a Z-basis for the homology of C(C). This basis will have four elements. Create a 2 × 4 complex matrix by integrating the two differentials over the basis for the homology. Create a 2 × 4 real matrix by taking the traces of entries from the complex matrix. The columns generate a lattice in R^{2}. The area of a fundamental domain is Ω.

It is worth mentioning an alternate strategy. Instead of finding a Z-basis for the homology of C(C) one could find a Z-basis for the subgroup of the homology which is fixed by complex conjugation (call this the real homology). This new basis would have two elements, and integrating would give us a 2 × 2 real matrix. The determinant of this matrix equals the integral of a generator of the integral 2-forms on J over the connected component of the real points of J. In other words, the index of the C/R-traces in the real homology is equal to the index of the connected component in J(R).

The curves in Table 1 are given as y^{2} + g(x)y = f(x). We assumed, at first, that dx/(2y + g(x)) and xdx/(2y+g(x)) generate the integral differentials. We integrated these differentials around each of the four paths generating the complex homology and found a provisional Ω. Then we checked the minimal, proper, regular models to determine if these differentials really did generate the integral differentials and adjusted the Ω when necessary. A result of Liu [Li, §4.1] shows that we need not check the differentials on genus 0 components of the special fiber with self-intersection number −2. This saved a lot of time as most components are of this type in our examples.

We were able to find generators of the integral differentials using arguments like those in the following example and the results of [Li, §5]. There were three curves where we needed to adjust Ω. We describe the adjustment for curve C_{65,B} in the following example. For curve C_{63}, we used the differentials 3dx/(2y + g(x)) and xdx/(2y + g(x)). For curve C_{65,A}, we used the differentials 3dx/(2y + g(x)) and 3xdx/(2y + g(x)). Example 3. Consider the curve

C_{65,B}. The primes of singular reduction of the curve are 2, 5 and 13. In Example 1 of section 3.5, we found that to get from the equation of the curve listed in Table 1 to a minimal, proper, regular model, we need to make the substitutions

x = \frac{αx_{1} − 2α + 2}{x_{1} − 2}, y = \frac{4y_{2} + 2(α − 1)x_{1}^{2}}{(x_{1} − 2)^{3}} .

Thus we have

\frac{dx}{2y} = \frac{(2 − x_{1})dx_{1}}{4y_{2} + 2(α − 1)x_{1}^{2}} , \frac{xdx}{2y} = \frac{(−αx_{1} + 2α − 2)dx_{1}}{4y_{2} + 2(α − 1)x_{1}^{2}} .

Now for the curve

y_{2}^{2} + (α − 1)x_{1}^{2}y_{2} = (−3α − 2)x_{1}^{5} + (27α + 19)x_{1}^{4} + (−112α − 46)x_{1}^{3}
+ (263α − 79)x_{1}^{2} + (−352α + 424)x_{1} + 208α − 416

we have the following two generators for the holomorphic differentials which are integral at 2:

A = \frac{dx_{1}}{2y_{2} + (α − 1)x_{1}^{2}} , B = \frac{x_{1}dx_{1}}{2y_{2} + (α − 1)x_{1}^{2}} .

We have

[ \begin{matrix}1 & −1/2 \\ α − 1 & −α/2 \end{matrix} ] [ \begin{matrix}A \\ B \end{matrix} ] = [ \frac{dx}{2y} ; \frac{xdx}{2y} ] .

The determinant of the matrix displayed is −1/2. Therefore we must multiply our provisional Ω by 2.

Now we must make sure that A and B are globally defined on the minimal, regular model. We have chosen differentials which are holomorphic on the generic fiber, so there is no problem there. Checking the special fiber we find they are both defined along A, but that this is not the case along D. One affine cover of the minimal model was g_{6}(x_{3}, y_{6}, p_{4}) = 0, x_{3}p_{4} = 2. We have x_{1} = 2x_{3} and y_{2} = 2x_{3}(y_{6}x_{3} − (α + 1)p_{4} − α). Thus we have

A = \frac{−dp_{4}}{2[2(y_{6} + α − 1) − (α + 1)p_{4}^{2} − αp_{4}]} B = \frac{−x_{3}dp_{4}}{2(y_{6} + α − 1) − (α + 1)p_{4}^{2} − αp_{4}} .

Note that along D, p_{4} − t is a uniformizer at p_{4} = t almost everywhere. Since it is clear that the denominator of B is almost everywhere non-zero, B is defined on D. When we check A along D we see that we must multiply it by 2 in order for the denominator to be non-zero. Thus we multiply our current provisional Ω by another 2 and get 4 times the original estimate for Ω.

Obtaining the minimal, proper, regular models of this curve at the primes 5 and 13 involved only shifting x-coordinates and so there was no change in Ω from these primes.

N  Equation  Source
23  y^{2} + (x^{3} + x + 1)y = x^{5} − 5x^{4} − 36x^{3} − 88x^{2} − 99x − 44  HW
29  y^{2} + (x^{3} + x^{2} + x)y = −5x^{4} − 17x^{3} − 21x^{2} − 8x − 1  HW
31  y^{2} + (x^{3} + x + 1)y = x^{5} − 4x^{4} − 24x^{3} − 40x^{2} − 25x − 7  HW
35  y^{2} + (x^{3} + x)y = −x^{5} − 8x^{3} − 7x^{2} − 16x − 19  H
39  y^{2} + (x^{3} + 1)y = −5x^{4} − 2x^{3} + 16x^{2} − 12x + 2  H
63  y^{2} + (x^{3} − 1)y = 14x^{3} − 7  W
65, A y^{2} + (x^{3} + x^{2} + x)y = −4x^{6} + 22x^{5} − 48x^{4} + 46x^{3} − 7x^{2} − 27x + 6  W
65, B  y^{2} = −x^{6} + 10x^{5} − 32x^{4} + 20x^{3} + 40x^{2} + 6x − 1  W
67  y^{2} + (x^{3} + x + 1)y = 2x^{5} − 5x^{4} + 3x^{3} − x^{2} − x  HW
73  y^{2} + (x^{3} + x^{2} + 1)y = −3x^{5} − 4x^{4} − x^{3} + x^{2} + x  HW
85  y^{2} + (x^{3} + 1)y = −x^{5} + 3x^{4} − 6x^{3} + 8x^{2} − 10x + 6  H
87  y^{2} + (x^{3} + x^{2} + 1)y = x^{5} + 3x^{4} + 4x^{3} + 2x^{2} − 1  HW
93  y^{2} + (x^{3} + x^{2} + 1)y = −2x^{5} + x^{4} + x^{3}  HW
103  y^{2} + (x^{3} + x^{2} + 1)y = x^{5} − 5x^{4} + 5x^{3} − 3x^{2}  HW
107  y^{2} + (x^{3} + x + 1)y = −x^{5} + 2x^{4} − 5x^{3} + 4x^{2} − 3x  HW
115  y^{2} + (5x^{3} + x^{2} + 1)y = 20x^{5} + 34x^{4} + 25x^{3} + 12x^{2} + 3x  HW
117, A  y^{2} + (x^{3} − 1)y = 3x^{3} − 7  W
117, B  y^{2} + (x^{3} + 1)y = −x^{6} − 3x^{4} − 5x^{3} − 12x^{2} − 9x − 7  W
125, A y^{2} + (x^{3} + x + 1)y = x^{5} − 3x^{4} − 23x^{3} − 54x^{2} − 58x − 24  HW
125, B y^{2} + (x^{3} + x + 1)y = x^{6} + 5x^{5} − 13x^{4} − 113x^{3} − 269x^{2} − 288x − 119 W
133, A y^{2} + (x^{3} + x + 1)y = −x^{6} + 10x^{5} − 48x^{4} + 92x^{3} − 63x^{2} + 17x − 2  W
133, B y^{2} + (x^{3} + x + 1)y = −x^{5} − 5x^{4} − 7x^{3} − 4x^{2} − x  HW
135 y^{2} + (x^{3} + x + 1)y = 3x^{5} + 16x^{4} + 49x^{3} + 83x^{2} + 67x + 11 W
147  y^{2} + (x^{3} + x^{2} + x)y = 7x^{5} + 92x^{4} + 605x^{3} + 2203x^{2} + 4242x + 3381 HW
161  y^{2} + (x^{3} + x + 1)y = −3x^{5} + 6x^{4} − 6x^{3} + 4x^{2} − 2x  HW
165  y^{2} + (x^{3} + 1)y = −x^{4} + 2x^{3} − 4x^{2} + 2  H
167  y^{2} + (x^{3} + x^{2} + 1)y = −3x^{5} + 9x^{4} − 18x^{3} + 17x^{2} − 8x  HW
175  y^{2} + (x^{3} + x + 1)y = x^{5} − 5x^{3} − 14x^{2} − 18x − 9  W
177  y^{2} + (x^{3} + x^{2} + 1)y = x^{5} + x^{4} + x^{3}  HW
188  y^{2} = x^{5} − x^{4} + x^{3} + x^{2} − 2x + 1  W
189  y^{2} + (x^{3} − 1)y = x^{3} − 7  W
191  y^{2} + (x^{3} + x^{2} + 1)y = −5x^{5} + 31x^{4} − 101x^{3} + 126x^{2} + 36x − 152  HW

Table 1. Levels, integral models and sources for curves

4. Tables

In Table 1, we list the 32 curves described in section 2. We give the level N from which each curve arose, an integral model for the curve, and list the source(s) from which it came (H for Hasegawa [Ha], W for Wang [Wa]). Throughout the paper, the curves are denoted C_{N}.

In Table 2, we list the curve C_{N} simply by N, the level from which it arose. Let r denote the analytic and Mordell-Weil ranks (they were always the same). We list lim_{s→1}(s−1)^{−r}L(J, s) where L(J, s) is the L-series for the Jacobian J of C_{N} and round off the results to five digits. The symbol Ω was defined in section 3.6 and is also rounded to five digits. Let Reg denote

N  r lim_{s→1} \frac{L(J,s)}{(s−1)^{r}} Ω  Reg  c_{p}’s Φ � ? error
23  0 0.24843 2.7328 1  11 11 1  < 10^{−120}
29  0 0.29152 2.0407 1  7  7 1  < 10^{−50}
31  0 0.44929 2.2464 1  5  5 1  < 10^{−49}
35  0 0.37275 2.9820 1  16,2 16 1  < 10^{−25}
39  0 0.38204 10.697 1  28,1 28 1  < 10^{−25}
63  0 0.75328 4.5197 1  2,3 6 1  < 10^{−49}
65,A 0 0.45207 6.3289 1  7,1 14 2  < 10^{−48}
65,B 0 0.91225 5.4735 1  1,3 6 2  < 10^{−50}
67  2 0.23410 20.465 0.011439 1  1 1  < 10^{−50}
73  2 0.25812 24.093 0.010713 1  1 1  < 10^{−49}
85  2 0.34334 9.1728 0.018715 4,2 2 1  < 10^{−26}
87  0 1.4323  7.1617 1  5,1 5 1  < 10^{−49}
93  2 0.33996 18.142 0.0046847 4,1 1 1  < 10^{−49}
103 2 0.37585 16.855 0.022299 1  1 1  < 10^{−49}
107 2 0.53438 11.883 0.044970 1  1 1  < 10^{−49}
115 2 0.41693 10.678 0.0097618 4,1 1 1  < 10^{−50}
117,A 0 1.0985  3.2954 1  4,3 6 1  < 10^{−49}
117,B 0 1.9510  1.9510 1  4,1 2 1  < 10^{−49}
125,A 2 0.62996 13.026 0.048361 1  1 1  < 10^{−50}
125,B 0 2.0842  2.6052 1  5  5 4  < 10^{−49}
133,A 0 2.2265  2.7832 1  5,1 5 4  < 10^{−49}
133,B 2 0.43884 15.318 0.028648 1,1 1 1  < 10^{−49}
135 0 1.5110 4.5331 1 3,1 3 1 < 10^{−49}
147 2 0.61816 13.616 0.045400 2,2 2 1  < 10^{−50}
161 2 0.82364 11.871 0.017345 4,1 1 1  < 10^{−47}
165 2 0.68650 9.5431 0.071936 4,2,2 4 1  < 10^{−26}
167 2 0.91530 7.3327 0.12482 1  1 1  < 10^{−47}
175 0 0.97209 4.8605 1  1,5 5 1  < 10^{−44}
177 2 0.90451 13.742 0.065821 1,1 1 1  < 10^{−45}
188 2 1.1708  11.519 0.011293 9,1 1 1  < 10^{−44}
189 0 1.2982  3.8946 1  1,3 3 1  < 10^{−43}
191 2 0.95958 17.357 0.055286 1  1 1  < 10^{−44}

Table 2. Conjectured sizes of � (J, Q)

the regulator, also rounded to five digits. We list the c_{p}’s by primes of increasing order dividing the level N. We denote J(Q)_{tors} = Φ and list its size. We use � ? to denote the size of (lim_{s→1}(s−1)^{−r}L(J, s))·(#J(Q)_{tors})^{2}/(Ω·Reg ·∏ c_{p}), rounded to the nearest integer. We will refer to this as the conjectured size of � (J, Q). The last column gives a bound on the rounding error.

Insert Joe’s paragraph on William’s verifications. In Table 3 are generators of J(Q)/J(Q)_{tors} for the curves whose Mordell-Weil ranks are 2.

There are two points at infinity in the normalization of the curves described by our equations,

N  Generators of J(Q)/J(Q)_{tors}
67  [(0, −1) + (1, −1) − ∞^{+} − ∞^{−}]  [(0, 0) + (0, 0) − ∞^{+} − ∞^{−}]
73  [(0, −1) + (−1, 0) − ∞^{+} − ∞^{−}]  [(0, 0) + (0, 0) − ∞^{+} − ∞^{−}]
85  [(0, 2) − ∞^{+}]  [(0, −3) − ∞^{+}]
93  [(1, 0) + (−1, −2) − ∞^{+} − ∞^{−}]  [(0, 0) + (1, 0) − ∞^{+} − ∞^{−}]
103 [(0, −1) + (1, −1) − ∞^{+} − ∞^{−}]  [(0, 0) + (0, 0) − ∞^{+} − ∞^{−}]]
107 [∞^{+} − ∞^{−}]  [(0, 0) − ∞^{−}]
115 [(−\frac{2}{3}, \frac{1}{9}) − ∞^{−}]  [(−\frac{1}{2} + \frac{1}{2}i, −1) + (−\frac{1}{2} − \frac{1}{2}i, −1) − ∞^{+} − ∞^{−}]
125,A [(−2, 4) − ∞^{−}]  [(−2, 5) + (−2, 5) − ∞^{+} − ∞^{−}]
133,B [(0, −1) + (−1, 1) − ∞^{+} − ∞^{−}]  [(0, 0) + (0, 0) − ∞^{+} − ∞^{−}]
147 [(−5, 51) + (−7, 147) − ∞^{+} − ∞^{−}] [(−4, 27) + (−7, 147) − ∞^{+} − ∞^{−}]
161 [(\frac{1}{2}, −\frac{3}{8}) − ∞^{+}]  [(\frac{1}{2}, −\frac{3}{8}) + (\frac{2}{3}, −\frac{1}{3}) − ∞^{+} − ∞^{−}]
165 [(1, −1) − ∞^{+}]  [(0, −2) − ∞^{+}]
167 [(0, 0) − ∞^{+}]  [(1 + i, 1 − 2i) + (1 − i, 1 + 2i) − ∞^{+} − ∞^{−}]
177 [(0, −1) − ∞^{−}]  [(0, 0) + (0, 0) − ∞^{+} − ∞^{−}]
188 [(0, 1) − ∞]  [(1, 1) − ∞]
191 [(4, −41) − ∞^{−}]  [(4, −40) + (4, −40) − ∞^{+} − ∞^{−}]

Table 3. Generators of J(Q)/J(Q)_{tors} in rank 2 cases

with the exception of curve C_{188}. The function y has a pole of higher order at one of them, which we will denote ∞^{−} and we will denote the other ∞^{+}.

In Table 4 are the reduction types, from the classification of [NU], of the special fibers of the minimal, proper, regular models of the curves for each of the primes of singular reduction for the curve. They are the same as the primes dividing the level except that curve C_{65,A} has singular reduction at the prime 3 and curve C_{65,B} has singular reduction at the prime 2.

5. Discussion of Tate-Shafarevich groups and evidence for the second conjecture

From section 3.3 we have dim � (J, Q)[2] = dim Sel^{2}(J, Q) − r − dim J(Q)[2]. With the exception of curves C_{65,A}, C_{65,B}, C_{125,B}, and C_{133,A} we have dim � (J, Q)[2] = 0. Thus we expect #� (J, Q) to be an odd square. In each case, the conjectured size of � (J, Q) is 1. For curves C_{65,A}, C_{65,B}, C_{125,B} and C_{133,A} we have dim � (J, Q)[2] = 1, 1, 2 and 2 and the conjectured size of � (J, Q) = 2, 2, 4 and 4, respectively. We see that in each case, the size of the odd part of � (J, Q) has size 1 and the 2-part is accounted for by its 2-torsion.

It is also interesting to consider deficient primes. A prime p is deficient with respect to a curve C of genus 2, if C has no degree 1 rational divisor over Q_{p}. From [PSt], the number of deficient primes has the same parity as dim � (J, Q)[2]. Curve C_{65,A} has one deficient prime 3. Curve C_{65,B} has one deficient prime 2. Curve C_{117,B} has two deficient primes 3 and ∞. The rest of the curves have no deficient primes. Recall, from section 3.3, we had proven that the Mordell-Weil rank, for the curves described by the given equations, was 0 in all cases when the analytic rank was 0, with the exception of curves C_{65,A}, C_{65,B}, C_{125,B} and C_{133,A}. These are among the 11 curves for which we can not prove that our equation is correct. Using the existence of an odd number of deficient primes for curves C_{65,A} and C_{65,B}, we can prove that their Mordell-Weil ranks are 0 also.

N  Prime Type  Prime Type N  Prime Type  Prime Type
23 23  I_{3−2−1}  117,A 3  III − III^{∗} − 0 13  I_{1−1−1}
29 29  I_{3−1−1}  117,B 3  I_{3−1−1}^{∗}  13  I_{1−1−0}
31 31  I_{2−1−1}  125,A 5  VIII − 1
35 5  I_{3−2−2}  7  I_{2−1−0} 125,B 5  IX − 3
39 3  I_{6−2−2}  13  I_{1−1−0} 133,A 7  I_{2−1−1}  19  I_{1−1−0}
63 3  2I_{0}^{∗} − 0  7  I_{1−1−1} 133,B 7  I_{1−1−0}  19  I_{1−1−0}
65,A 3  I_{0} − I_{0} − 1 5  I_{3−1−1} 135 3  III  5  I_{3−1−0}
65,A 13  I_{1−1−0}  147 3  I_{2−1−0}  7  VII
65,B 2  I_{0} − I_{0} − 1 5  I_{3−1−0} 161 7  I_{2−2−0}  23  I_{1−1−0}
65,B 13  I_{1−1−1}  165 3  I_{2−2−0}  5  I_{2−1−0}
67 67  I_{1−1−0}  165 11  I_{2−1−0}
73 73  I_{1−1−0}  167 167 I_{1−1−0}
85 5  I_{2−2−0}  17  I_{2−1−0} 175 5  II − II − 0 7  I_{2−1−1}
87 3  I_{2−1−1}  29  I_{1−1−0} 177 3  I_{1−1−0}  59  I_{1−1−0}
93 3  I_{2−2−0}  31  I_{1−1−0} 188 2  IV − IV − 0 47  I_{1−1−0}
103 103 I_{1−1−0}  189 3  II − IV^{∗} − 0 7  I_{1−1−1}
107 107 I_{1−1−0}  191 191 I_{1−1−0}
115 5 I_{2−2−0} 23 I_{1−1−0}

Table 4. Namikawa and Ueno classification of special fibers

6. Other Hasegawa curves

In Table 5 is data concerning all 142 of Hasegawa’s curves in the order presented in his paper. Let us explain the entries. The first column in each set of three columns gives the level, N. The second column gives a classification of the cusp forms spanning the 2-dimensional subspace of S_{2}(N) corresponding to the Jacobian. When that subspace is irreducible with respect to the action of the Hecke algebra and is spanned by two newforms or two oldforms, we write 2n or 2o, respectively. When that subspace is reducible and is spanned by two oldforms, two newforms or one of each, we write oo, nn and on, respectively. The third column contains the sign of the functional equation at the level M at which the cusp form is a newform. This is the negative of ϵ_{M} (described in section 3.1). The order of the two signs in the third column agrees with that of the forms listed in the second column. We include this information for those who would like to further study these curves. The curves with N < 200 classified as 2n appeared already in Table 1.

The minimal possible Mordell-Weil ranks corresponding to ++, +−, −+ and −−, predicted by the first Birch and Swinnerton-Dyer conjecture, are 0, 1, 1 and 2 respectively. In all cases, those were, in fact, the Mordell-Weil ranks. This was determined by computing 2-Selmer groups with a computer program based on [St2]. Of course, these are cases where the first Birch and Swinnerton-Dyer conjecture is already known to hold. In the cases where the Mordell-Weil rank is positive, the Mordell-Weil group has a subgroup of finite index generated by points with small rational x-coordinates with numerators bounded by 7 (in absolute value) and denominators by 12 with one exception. On the second curve with N = 138, the divisor class [(3 + 2\sqrt{2}, 80 + 56\sqrt{2}) + (3 − 2\sqrt{2}, 80 − 56\sqrt{2}) − 2∞] generates a subgroup of finite index in the Mordell-Weil group.

22 oo ++ 58 nn +− 87 2o ++ 129 on −− 198 2o +−
23 2n ++ 60 oo ++ 88 on +− 130 on −+ 204 2o +−
26 nn ++ 60 2o ++ 90 on ++ 132 oo ++ 205 2n −−
28 oo ++ 60 2o ++ 90 oo ++ 133 2n −− 206 2o −−
29 2n ++ 62 2o ++ 90 oo ++ 134 2o −− 209 2n −−
30 on ++ 66 nn ++ 90 oo ++ 135 on +− 210 on +−
30 oo ++ 66 2o ++ 91 nn −− 138 nn +− 213 2n −−
30 on ++ 66 2o ++ 93 2n −− 138 on +− 215 on −−
31 2n ++ 66 on ++ 98 oo ++ 140 oo ++ 221 2n −−
33 on ++ 67 2n −− 100 oo ++ 142 nn +− 230 2o −−
35 2n ++ 68 oo ++ 102 on +− 143 on +− 255 2o −−
37 nn +− 69 2o ++ 102 on +− 146 2o −− 266 2o −−
38 on ++ 70 on ++ 103 2n −− 147 2n −− 276 2o +−
39 2n ++ 70 2o ++ 104 2o ++ 150 on ++ 284 2o +−
40 on ++ 70 2o ++ 106 on −− 153 on +− 285 on −−
40 oo ++ 70 2o ++ 107 2n −− 154 on −− 286 on −−
42 on ++ 72 on ++ 110 on ++ 156 oo ++ 287 2n −−
42 oo ++ 72 oo ++ 111 oo +− 158 on −− 299 2n −−
42 on ++ 73 2n −− 112 on +− 161 2n −− 330 2o −−
42 oo ++ 74 oo +− 114 oo +− 165 2n −− 357 2n −−
44 2o ++ 77 on +− 115 2n −− 166 on −− 380 2o +−
46 2o ++ 78 oo ++ 116 2o +− 167 2n −− 390 on −−
48 on ++ 78 2o ++ 117 2o ++ 168 2o ++
48 oo ++ 80 oo ++ 120 oo ++ 170 2o −−
50 nn ++ 84 oo ++ 120 on ++ 177 2n −−
52 oo ++ 84 oo ++ 121 on +− 180 2o ++
52 oo ++ 84 oo ++ 122 on −− 184 on +−
54 on ++ 84 oo ++ 125 2n −− 186 2o −−
57 on +− 85 2n −− 126 oo ++ 190 on +−
57 on +− 87 2n ++ 126 on ++ 191 2n −−

Table 5. Spaces of cusp forms associated to Hasegawa’s curves

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Department of Mathematical Sciences, University of Liverpool, P.O.Box 147, Liverpool L69 3BX, England

  E-mail address: evflynn@liverpool.ac.uk

CNRS Equipe d’arithmétique, Institut de Mathématiques de Paris, Université Paris 6, Tour 46-56, 5ème étage, Case 247, 2-4 place Jussieu, F-75252 Paris cedex 05

  E-mail address: leprevot@math.jussieu.fr

Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, CA 95053, USA

  E-mail address: eschaefe@math.scu.edu

Mathematisches Institut, Universitätsstr. 1, D-40225 Düsseldorf, Germany

  E-mail address: stoll@math.uni-duesseldorf.de

Department of Mathematics, University of Southern California, 1042 W. 36th Place, Los Angeles, CA 90089-1113, USA

  E-mail address: jlwether@alum.mit.edu