The Modular Curve X_{0}(389):

Discriminants, Ranks, Shafarevich-Tate Groups, and Weierstrass Points

William Stein

April 2002

1 Introduction

Let N be a positive integer, and let X_{0}(N) be the compactified coarse moduli space that classifies pairs (E, C) where E is an elliptic curve and C is a cyclic subgroup of order N. The space X_{0}(N) has a canonical structure of algebraic curve over ℚ, and its properties have been very well studied during the last forty years. For example, Breuil, Conrad, Diamond, Taylor, and Wiles proved that every elliptic curve over ℚ is a quotient of some X_{0}(N).

The smallest N such that the Jacobian of X_{0}(N) has positive Mordell-Weil rank is 37, and Zagier studied the genus-two curve X_{0}(37) in depth in his paper [18]. From this viewpoint, the next modular curve deserving intensive investigation is X_{0}(389), which is the first modular curve whose Jacobian has Mordell-Weil rank larger than that predicted by the signs in the functional equations of the L-series attached to simple factors of its Jacobian; in fact, 389 is the smallest conductor of an elliptic curve with Mordell-Weil rank 2. Note that 389 is prime and X_{0}(389) has genus g = 32, which is much larger than the genus 2 of X_{0}(37), which makes explicit investigation more challenging.

Work of Kolyvagin [9, 8] and Gross-Zagier [7] has completely resolved the rank assertion of the Birch and Swinnerton-Dyer conjecture (see, e.g., [17]) for elliptic curves E with ord_{s=1} L(E, s) ≤ 1. The lowest-conductor elliptic curve E that doesn’t submit to the work of Kolyvagin and Gross-Zagier is the elliptic curve E of conductor 389 mentioned in the previous paragraph. At present we don’t even have a conjectural natural construction of a finite-index subgroup of E(ℚ) analogous to that given by Gross and Zagier for rank 1 (but see Mazur’s work on universal norms, which might be used to construct E(ℚ) ⊗ ℤ_{p} for some auxiliary prime p).

Inspired by the above observations, and with an eye towards providing helpful data for anyone trying to generalize the work of Gross, Zagier, and Kolyvagin, in this paper we compute everything we can about the modular curve X_{0}(389). Some of the computations of this paper have already proved important in several other papers: the discriminant of the Hecke algebra attached to X_{0}(389) plays a roll in [14], the verification of condition 3 in [11], and the remark after Theorem 1 of [10]; also, the arithmetic of J_{0}(389) provides a key example in [3, §4.2]. Finally, this paper serves as an entry in an “encyclopaedia, atlas or hiker’s guide to modular curves”, in the spirit of N. Elkies (see [5, pg. 22]).

We hilight several surprising “firsts” that occur at level 389. The discriminant of the Hecke algebra attached to S_{2}(Γ_{0}(389)) has the apparently unusual property that it is divisible by p = 389 (see Section 3.1). Also N = 389 is the smallest integer such that the order of vanishing of L(J_{0}(N), s) at s = 1 is larger than predicted by the functional equations of eigenforms (see Section 2.3). The author conjectures that N = 389 is the smallest level such that an optimal newform factor of J_{0}(N) appears to have Shafarevich-Tate group with nontrivial odd part (see Section 5.1). Atkin conjectures that 389 is the largest prime such that the cusp of X_{0}^{+}(389) fails to be a Weierstrass point (see Section 5.2). Acknowledgement. Noam Elkies, Ken Ribet, Matt Baker. Hendrik Lenstra, for suggesting a method to compute discriminants of Hecke algebra efficiently (see Section 3).

2 Factors of J_{0}(389)

To each newform f ∈ S_{2}(Γ_{0}(389)), Shimura [16] associated a quotient A_{f} of J_{0}(389), and J_{0}(389) is isogeneous to the product ∏ A_{f}, where the product runs over the Gal(\overline{ℚ}/ℚ)-conjugacy classes of newforms. Moreover, because 389 is prime each factor A_{f} cannot be decomposed further up to isogeny, even over \overline{ℚ} (see [13]).

2.1 Newforms of Level 389

There are five Gal(\overline{ℚ}/ℚ)-conjugacy classes of newforms in S_{2}(Γ_{0}(389)). The first class corresponds to the unique elliptic curve of conductor 389, and its q-exansion begins

f_{1} = q − 2q^{2} − 2q^{3} + 2q^{4} − 3q^{5} + 4q^{6} − 5q^{7} + q^{9} + 6q^{10} + · · · .

The second has coefficients in the quadratic field ℚ(\sqrt{2}), and has q-expansion

f_{2} = q + \sqrt{2}q^{2} + (\sqrt{2} − 2)q^{3} − q^{5} + (−2\sqrt{2} + 2)q^{6} + · · · .

The third has coefficients in the cubic field generated by a root α of x^{3} − 4x − 2:

f_{3} = q + αq^{2} − αq^{3} + (α^{2} − 2)q^{4} + (−α^{2} + 1)q^{5} − α^{2}q^{6} + · · · .

The fourth has coefficients that generate the degree-six field defined by a root β of x^{6} + 3x^{5} − 2x^{4} − 8x^{3} + 2x^{2} + 4x − 1 and q-expansion

f_{4} = q + βq^{2} + (β^{5} + 3β^{4} − 2β^{3} − 8β^{2} + β + 2)q^{3} + · · · .

PSfrag replacements f_{1} f_{2} f_{3} f_{4} f_{5} 2 2 2, 5 3 31 2 2 2 2

Figure 2.1: Congruences Between Newforms

The fifth and final newform (up to conjugacy) has coefficients that generate the degree 20 field defined by a root of

f_{5} = x^{20} − 3x^{19} − 29x^{18} + 91x^{17} + 338x^{16} − 1130x^{15} − 2023x^{14} + 7432x^{13}
+6558x^{12} − 28021x^{11} − 10909x^{10} + 61267x^{9} + 6954x^{8} − 74752x^{7}
+1407x^{6} + 46330x^{5} − 1087x^{4} − 12558x^{3} − 942x^{2} + 960x + 148.

2.1.1 Congruences

The vertices in Figure 2.1 correspond to the newforms f_{i}; there is an edge between f_{i} and f_{j} labeled p if there is a maximal ideal ℘ | p of the field generated by the Fourier coefficients of f_{i} and f_{j} such that f_{i} ≡ f_{j} (mod ℘).

2.2 Isogeny Structure

We deduce from the above determination of the newforms in S_{2}(Γ_{0}(389)) that J_{0}(389) is ℚ-isogenous to a product of \overline{ℚ}-simple abelian varieties

J ∼ A_{1} × A_{2} × A_{3} × A_{4} × A_{5}.

View the duals A_{i}^{∨} of the A_{i} as abelian subvarieties of J_{0}(389). Using modular symbols as in [2, §3.4] we find that, for i ≠ j, a prime p divides #(A_{i}^{∨} ∩ A_{j}^{∨}) if and only if f_{i} ≡ f_{j} (mod ℘) for some prime ℘ | p (recall that the congruence primes are given in Figure 2.1 above).

2.3 Mordell-Weil Ranks

Suppose f ∈ S_{2}(Γ_{0}(N)) is a newform of some level N. The functional equation for L(f, s) implies that ord_{s=1} L(f, s) is odd if and only if the sign of the eigenvalue of the Atkin-Lehner involution W_{N} on f is +1.

Proposition 2.1. If f ∈ S_{2}(Γ_{0}(N)) is a newform of level N < 389, then ord_{s=1} L(f, s) is either0 or 1.

Proof. The proof amounts to a large computation, which divides into two parts:

1. Verify, for each newform f of level N < 389 such that W_{N}(f) = −f, that L(f, 1) = ∗ ∫_{0}^{i∞} f(z)dz (for some nonzero ∗) is nonzero. This is a purely algebraic computation involving modular symbols.

2. Verify, for each newform f of level N < 389 such that W_{N}(f) = f, that L′(f, 1) ≠ 0 (see [6, §4.1], which points to [4, §2.11,§2.13]). We do this by approximating an infinite series that converges to L′(f, 1) and noting that the value we get is far from 0.

Thus N = 389 is the smallest level such that the L-series of some factor A_{f} of J_{0}(N) has order of vanishing higher than that which is forced by the sign in the functional equation.

Proposition 2.2. The following table summarizes the dimensions and Mordell-Weil ranks (over the image of the Hecke ring) of the newform factors of J_{0}(N):

A_{1} A_{2} A_{3} A_{4} A_{5}
Dimension 1 2 3 6 20
Rank  2 1 1 1 0

Proof. The elliptic curve A_{1} is 389A in Cremona’s tables, which is the elliptic curve of smallest conductor having rank 2. For A_{5} we directly compute whether or not the L-function vanishes using modular symbols, by taking an inner product with the winding element e_{w} = −{0, ∞}. We find that the L-function does not vanish. By Kolyvagin-Logachev, it follows that A_{5} has Mordell-Weil rank 0.

For each of the other three factors, the sign of the functional equation is odd, so the analytic ranks are odd. As in the proof of Proposition 2.1, we verify that the analytic rank is 1 in each case. By work of Gross, Zagier, and Kolyvagin it follows that the ranks are 1.

3 The Hecke algebra

3.1 The Discriminant is Divisible By p

Let N be a positive integer. The Hecke algebra T ⊂ End(S_{2}(Γ_{0}(N))) is the subring generated by all Hecke operators T_{n} for n = 1, 2, 3, . . .. We are concerned with the discriminant of the trace pairing (t, s) ↦ Tr(ts).

When N is prime, T_{ℚ} = T ⊗_{ℤ} ℚ is a product K_{1} × ⋯ × K_{n} of totally real number fields. Let T̃ denote the integral closure of T in T_{ℚ}; note that T̃ = ∏ O_{i} where O_{i} is the ring of integers of K_{i}. Then disc(T) = [T̃ : T] · ∏_{i=1}^{n} disc(K_{i}).

Proposition 3.1. The discriminant of the Hecke algebra associated to S_{2}(Γ_{0}(389)) is

2^{53} · 3^{4} · 5^{6} · 31^{2} · 37 · 389 · 3881 · 215517113148241 · 477439237737571441.

Proof. By [1], the Hecke algebra T is generated as a ℤ-module by T_{1}, T_{2}, … T_{65}. To compute disc(T), we proceed as follows. First, compute the space S_{2}(Γ_{0}(389)) of cuspidal modular symbols, which is a faithful T-module. Choose a random element x ∈ S_{2}(Γ_{0}(389))_{+} of the +1-quotient of the cuspidal modular symbols, then compute the images v_{1} = T_{1}(x), v_{2} = T_{2}(x), …, v_{65} = T_{65}(x). If these don’t span a space of dimension 32 = rank_{ℤ} T choose a new random element x and repeat. Using the Hermite Normal Form, find a ℤ-basis b_{1}, …, b_{32} for the ℤ-span of v_{1}, …, v_{65}. The trace pairing on T induces a trace pairing on the v_{i}, and hence on the b_{i}. Then disc(T) is the discriminant of this pairing on the b_{i}. The reason we embed T in S_{2}(Γ_{0}(389))_{+} as Tx is because directly finding a ℤ-basis for T would involve computing the Hermite Norm Form of a list of 65 vectors in a 1024-dimensional space, which is unnecessarily difficult (though possible).

We compute this discriminant by applying the definition of discriminant to a matrix representation of the first 65 Hecke operators T_{1}, …, T_{65}. Matrices representing these Hecke operators were computed using the modular symbols algorithms described in [4]. [Sturm, On the congruence of modular forms].

In the case of X_{0}(389), T ⊗ ℚ = K_{1} × K_{2} × K_{3} × K_{6} × K_{20}, where K_{d} has degree d over ℚ. We have

K_{1} = ℚ,
K_{2} = ℚ(\sqrt{2})
K_{3} = ℚ(β), β^{3} − 4β − 2 = 0,
K_{6} = ℚ(γ), γ^{6} + 3γ^{5} − 2γ^{4} − 8γ^{3} + 2γ^{2} + 4γ − 1 = 0,
K_{20} = ℚ(δ), δ^{20} − 3δ^{19} − 29δ^{18} + 91δ^{17} + 338δ^{16} − 1130δ^{15} − 2023δ^{14} + 7432δ^{13}
+6558δ^{12} − 28021δ^{11} − 10909δ^{10} + 61267δ^{9} + 6954δ^{8} − 74752δ^{7}

+1407δ^{6} + 46330δ^{5} − 1087δ^{4} − 12558δ^{3} − 942δ^{2} + 960δ + 148 = 0.

The discriminants of the K_{i} are

\begin{matrix}K_{1} & K_{2} & K_{3} & K_{6} \\ 1 & 2^{3} & 2^{2} · 37 & 5^{3} · 3881 \end{matrix}

and

disc(K_{20}) = 2^{14} · 5 · 389 · 215517113148241 · 477439237737571441.

Observe that the discriminant of K_{20} is divisible by 389. The product of the discriminants is

2^{19} · 5^{4} · 37 · 389 · 3881 · 215517113148241 · 477439237737571441.

This differs from the exact discriminant by a factor of 2^{34} ·3^{4} ·5^{2} ·31^{2}, so the index of T in its normalization is

[T̃ : T] = 2^{17} · 3^{2} · 5 · 31.

Notice that 389 does not divide this index, and that 389 is not a “congruence prime”, so 389 does not divide any modular degrees.

Question 3.2. Is there a newform optimal quotient A_{f} of J_{0}(p) such that p divides the modular degree of A_{f}? (No, if p < 14000.)

3.2 Congruences Primes in S_{p+1}(Γ_{0}(1))

K. Ono asked the following question, in connection with Theorem 1 of [10].

Question 3.3. Let p be a prime. Is p ever a congruence prime on S_{p+1}(Γ_{0}(1))? More precisely, if K is the number field generated by all the eigenforms of weight p + 1 on Γ_{0}(1), can there be a prime ideal ℘ | p for which f ≡ g (mod ℘) for distinct eigenforms f, g ∈ S_{p+1}(Γ_{0}(1))?

The answer is “yes”. There is a standard relationship between S_{p+1}(Γ_{0}(1)) and S_{2}(Γ_{0}(p)). As noted in Section 3.1, p = 389 is a congruence prime for S_{2}(Γ_{0}(389)), so we investigate S_{389+1}(Γ_{0}(1)).

Proposition 3.4. There exist distinct newforms f, g ∈ S_{389+1}(Γ_{0}(1)) and a prime ℘ of residue characteristic 389 such that f ≡ g (mod ℘).

Proof. We compute the characteristic polynomial f of the Hecke operator T_{2} on S_{389+1}(Γ_{0}(1)) using nothing more than [15, Ch. VII]. We find that f factors modulo 389 as follows:

\overline{f} = (x + 2)(x + 56)(x + 135)(x + 158)(x + 175)^{2}(x + 315)(x + 342)(x^{2} + 387)
(x^{2} + 97x + 164)(x^{2} + 231x + 64)(x^{2} + 286x + 63)
(x^{5} + 88x^{4} + 196x^{3} + 113x^{2} + 168x + 349)
(x^{11} + 276x^{10} + 182x^{9} + 13x^{8} + 298x^{7} + 316x^{6} + 213x^{5}
+248x^{4} + 108x^{3} + 283x^{2} + x + 101)

Moreover, f is irreducible and 389 || disc(f), so the square factor (x+175)^{2} implies that 389 is ramified in the degree-32 field L generated by a single root of f. Thus there are exactly 31 distinct homomorphisms from the ring of integers of L to \overline{𝔽}_{389}. That is, there are exactly 31 ways to reduce the q-expansion of a newform in S_{390}(Γ_{0}(1)) to obtain a q-expansion in \overline{𝔽}_{389}[[q]]. Let K be the field generated by all eigenvalues of the 32 newforms g_{1}, … g_{32} ∈ S_{390}(Γ_{0}(1)), and let ℘ be a prime of O_{K} lying over 389. Then the subset {g_{1} (mod ℘), g_{2} (mod ℘), …, g_{32} (mod ℘)} of \overline{𝔽}_{389}[[q]] has cardinality at most 31, so there exists i ≠ j such that g_{i} ≡ g_{j} (mod ℘).

4 Supersingular Points in Characteristic 389

4.1 The Supersingular j-invariants In Characteristic 389

Let α be a root of α^{2} + 95α + 20. Then the 33 = g(X_{0}(389)) + 1 supersingular j-invariants in 𝔽_{389^{2}} are

0, 7, 16, 17, 36, 121, 154, 220, 318, 327, 358, 60α + 22, 68α + 166, 80α + 91, 86α + 273, 93α + 333, 123α + 350, 123α + 375, 129α + 247, 131α + 151, 160α + 321, 176α + 188, 213α + 195, 229α + 292, 258α + 154, 260α + 51, 266α + 335, 266α + 360, 296α + 56, 303α + 272, 309α + 271, 321α + 319, 329α + 157.

Figure 4.1 contains the graph of the Hecke operator T_{2} computed using method of Mestre and Oesterlé [12]. Here, if T_{2}(j) = ∑_{i} j_{i}, then there is an edge from the node j to each of the nodes j_{i}. Thus, for example,

T_{2}([318]) = [0] + [93α + 333] + [296α + 56].

4.2 Gross Triple-Product Business

5 Miscellaneous

5.1 The Shafarevich-Tate Group

Using visibility theory [3, §4.2], one sees that #� (A_{5}) is divisible by an odd prime, because

(ℤ/5ℤ)^{2} ≈ A_{1}(ℚ)/5A_{1}(ℚ) ⊂ � (A_{5}).

Additional computations suggest the following conjecture.

Conjecture 5.1. N = 389 is the smallest level such that there is an optimal newform quotient A_{f} of J_{0}(N) with #� (A_{f}) divisible by an odd prime.

PSfrag replacements

0 7 16 17 36 121 154 220 318 327 358

60α + 22 68α + 166 80α + 91 86α + 273 93α + 333 123α + 350 123α + 375 129α + 247 131α + 151 160α + 321 176α + 188 213α + 195 229α + 292 258α + 154 260α + 51 266α + 335 266α + 360 296α + 56 303α + 272 309α + 271 321α + 319 329α + 157

Figure 4.1: Graph of the Hecke Operator T_{2}

5.2 Weierstrass Points on X_{0}^{+}(p)

Oliver Atkin has conjectured that 389 is the largest prime such that the cusp on X_{0}^{+}(389) fails to be a Weierstrass point. He verified that the cusp of X_{0}^{+}(389) is not a Weierstrass point but that the cusp of X_{0}^{+}(p) is a Weierstrass point for all primes p such that 389 < p ≤ 883 (see, e.g., [5, pg.39]). In addition, the author has extended the verification of Atkin’s conjecture for all primes < 3000. Explicitly, this involves computing a reduced-echelon basis for the subspace of S_{2}(Γ_{0}(p)) where the Atkin-Lehner involution W_{p} acts as +1, and comparing the largest valuation of an element of this basis with the dimension of the subspace. These numbers differ exactly when the cusp is a Weierstrass point.

5.3 A Property of the Plus Part of the Integral Homology

For any positive integer N, let H^{+}(N) = H_{1}(X_{0}(N), ℤ)^{+} be the +1 eigen-submodule for the action of complex conjugation on the integral homology of X_{0}(N). Then H^{+}(N) is a module over the Hecke algebra T. Let

F^{+}(N) = coker (H^{+}(N) × Hom(H^{+}(N), ℤ) → Hom(T, ℤ))

where the map sends (x, φ) to the homomorphism t ↦ φ(tx). Then #F^{+}(p) ∈ {1, 2, 4} for all primes p < 389, but #F^{+}(389) = 8.

5.4 The Field Generated By Points of Small Prime Order On an Elliptic Curve

The prime 389 arises in a key way in the verification of condition 3 in [11].

References

[1] A. Agashe and W. A. Stein, Appendix to Joan-C. Lario and René Schoof: Some computations with Hecke rings and deformation rings, submitted.

[2] , Visible evidence for the birch and swinnerton-dyer conjecture for modular abelian varieties of analytic rank 0, In Preparation.

[3] , Visibility of Shafarevich-Tate Groups of Abelian Varieties, to appear in J. of Number Theory (2002).

[4] J. E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997.

[5] N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, Computational perspectives on number theory (Chicago, IL, 1995), Amer. Math. Soc., Providence, RI, 1998, pp. 21–76.

[6] E. V. Flynn, F. Leprévost, E. F. Schaefer, W. A. Stein, M. Stoll, and J. L. Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comp. 70 (2001), no. 236, 1675–1697 (electronic).

[7] B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320.

[8] V. A. Kolyvagin, Finiteness of E(q) and SH(E, q) for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522–540, 670–671.

[9] , The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1154–1180, 1327.

[10] W. J. McGraw and K. Ono, Modular form Congruences and Selmer groups, preprint (2002).

[11] L. Merel and W. A. Stein, The field generated by the points of small prime order on an elliptic curve, Internat. Math. Res. Notices (2001), no. 20, 1075– 1082.

[12] J.-F. Mestre, La méthode des graphes. Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata) (1986), 217–242.

[13] K. A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. Math. (2) 101 (1975), 555–562.

[14] , Torsion points on J_{0}(N) and Galois representations, Arithmetic theory of elliptic curves (Cetraro, 1997), Springer, Berlin, 1999, pp. 145–166.

[15] J-P. Serre, A Course in Arithmetic, Springer-Verlag, New York, 1973, Translated from the French, Graduate Texts in Mathematics, No. 7.

[16] G. Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), no. 3, 523–544.

[17] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 306, 415–440.

[18] D. Zagier, Modular points, modular curves, modular surfaces and modular forms, Workshop Bonn 1984 (Bonn, 1984), Springer, Berlin, 1985, pp. 225– 248.