SYMMETRIC SQUARES OF ELLIPTIC CURVES: RATIONAL POINTS AND SELMER GROUPS

NEIL DUMMIGAN

1. Introduction

Let E be an elliptic curve defined over ℚ and let L(E, s) be the associated L-function. (It is now known that E is modular, so that L(E, s) has an analytic continuation to the whole complex plane.) The conjecture of Birch and Swinnerton-Dyer predicts that the order of vanishing of L(E, s) at s = 1 is the rank of the group E(ℚ) of rational points, and also gives an interpretation of the leading term in the Taylor expansion in terms of various quantities, including the order of the Shafarevich-Tate group.

The Bloch-Kato conjecture [BK] is the generalisation to arbitrary motives of the leading term part of the Birch and Swinnerton-Dyer conjecture. Let L(Sym^{2}(E), s) be the symmetric square L-function attached to an elliptic curve E/ℚ. Flach [Fl1] looked at the Bloch-Kato conjecture for L(Sym^{2}(E), s) at s = 2, and translated it into a formula involving only rational numbers, such as the degree of a modular parametrisation, and the order of a generalised Shafarevich-Tate group. In [Fl2] he applied Kolyvagin’s technique for bounding Selmer groups, to Sym^{2}(E) at s = 2. Theorem 1 of [Fl1] applies the result of [Fl2] to prove the l-part of the Bloch-Kato formula for all primes l > 3 such that E has good reduction at l, l does not divide the degree of the modular parametrisation, and the representation Gal(\overline{ℚ}/ℚ) → GL_{2}(𝔽_{l}) arising from the action of Gal(\overline{ℚ}/ℚ) on the l-torsion points of E is surjective. (Under these conditions, the l-part of the generalised Shafarevich-Tate group is trivial.)

Modified l-Selmer groups associated to Sym^{2}(E) at s = 2 are intimately connected with the deformation theory of the above Galois representation. (More generally, E may be replaced by a modular form of weight 2, with not necessarily rational coefficients.) This connection lies at the heart of Wiles’s approach to the Shimura-Taniyama-Weil conjecture, that every elliptic curve E/ℚ is modular. Although the work of Taylor and Wiles [Wi], [TW] does not actually prove the relevant cases of the Bloch-Kato conjecture (the Selmer groups are defined differently), it is clearly closely related.

Diamond, Flach and Guo [DFG1], [DFG2] have now proved a general result on the Bloch-Kato conjecture (at s = k) for the symmetric square of a newform of weight k ≥ 2. Applying their result to L(Sym^{2}(E), 2) proves the l-part of the Bloch-Kato conjecture for primes l > 2 where E has good reduction and the representation of Gal(\overline{ℚ}/ℚ(\sqrt{(−1)^{(l−1)/2}l})) on E[l] is absolutely irreducible.

Date: June 15th, 2001.

1991 Mathematics Subject Classification. 11F33, 11F37, 11F67, 11G40.

Key words and phrases. elliptic curve, symmetric square L-function, Bloch-Kato conjecture.

In [Du1] I looked at L(Sym^{2}(f), s), where f is a Hecke eigenform of level one and weight k, concentrating on the weights k = 12, 16, 18, 20, 22, 26, for which f has rational Fourier coefficients. For k = 18, 22, 26 (when k/2 is odd) the Bloch-Kato conjecture turns out to be especially interesting at the point s = (k − 1) + (k/2). It is possible to construct elements of order l in the relevant Selmer groups, where l is an “Eisenstein” prime, and these primes do appear, at s = (k−1)+(k/2), when the critical values of the L-function are calculated. [Du2] deals with something similar for a Hilbert modular form.

The point s = (k − 1) + (k/2) coincides with the point s = k dealt with in [DFG1] only when k = 2. (Of course, if k = 2 there are no non-zero modular forms of level one, but the level one restriction is no longer necessary when we have an elliptic curve to work with.) The construction of [Du1] can be made to work in the case that E(ℚ) has both a point of order l and a point of infinite order. The point of infinite order gives rise, via the “descent” map, to a non-zero element of H^{1}(ℚ, E[l]). Thanks to the existence of the rational point of order l, E[l] is isomorphic to a Galois submodule of Sym^{2}(E[l]), and we get a non-zero element c ∈ H^{1}(ℚ, Sym^{2}(E[l])). The main problem we face is to show that c (or rather its image in another group) satisfies all the local conditions required for it to belong to the appropriate Bloch-Kato Selmer group. To facilitate this, we assume that E has good reduction at l, and impose some technical conditions which are satisfied by most of the examples we look at (see the precise statement of Theorem 5.1).

In [CM], Cremona and Mazur look, among all strong Weil elliptic curves over ℚ of conductor N ≤ 5500, at those with non-trivial Shafarevich-Tate group (according to the Birch and Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate group has predicted elements of order m. In most cases they find another elliptic curve, often of the same conductor, whose m-torsion is Galois-isomorphic to that of the first one, and which has rank two. The rational points on the second elliptic curve produce classes in the common H^{1}(ℚ, E[m]). They expect that these lie in the Shafarevich-Tate group of the first curve, so rational points on one curve explain elements of the Shafarevich-Tate group of the other curve. Clearly, the construction of the present paper is analogous to this.

Ironically, the rational point of order l which allows the construction to proceed causes the representation of Gal(\overline{ℚ}/ℚ) to be reducible, so the results of [Fl2] and [DFG1] do not apply to the l-part of the Bloch-Kato conjecture for L(Sym^{2}(E), 2). Therefore it is appropriate to consider numerical evidence. We must take l > 3, since the 2- and 3-parts of the fudge factors occurring in this instance of the Bloch-Kato formula are not well understood. Since there cannot be a rational l-torsion point for l > 7 [M], the only relevant l are l = 5 and l = 7. We concentrate on the case l = 5, which occurs much more frequently. With the exception of the order of the (generalised) Shafarevich-Tate group, all the quantities appearing in the l-part of the Bloch-Kato formula appear in, or may be calculated from, the elliptic curve data in Cremona’s tables, for all conductors N ≤ 7000. These tables may be found at http://www.maths.nott.ac.uk/ personal/jec/ftp/data/INDEX.html. We find that the data fits well with both the conjecture and our construction.

I am grateful to J. Cremona, for making his data public and for double-checking one of the modular parametrisation degrees for me, and to M. Flach for correcting his Lemma 1 in [Fl1], and for informing me of [DFG1].

2. The symmetric square L-function

Let E/ℚ be an elliptic curve, of conductor N. Let l be a prime number, let T_{l} = \underleftarrow{lim} E[l^{n}] be the l-adic Tate module of E, and V_{l} = T_{l} ⊗ ℚ_{l}. Let A_{l} = V_{l}/T_{l} = ∪_{n=1}^{∞}E[l^{n}]. The absolute Galois group Gal(\overline{ℚ}/ℚ) acts continuously on all of these modules, in a natural way. As Galois modules, V_{l} ≃ H^{1}(\overline{E}, ℚ_{l})(1).

Let V′_{l} = Sym^{2}(V_{l}), T′_{l} = Sym^{2}(T_{l}) and A′_{l} = V′_{l}/T′_{l} = ∪_{n=1}^{∞}A′[l^{n}], where A′[l^{n}] := Sym^{2}(E[l^{n}]). As Galois modules, V′_{l} ≃ Sym^{2}(H^{1}(\overline{E}, ℚ_{l}))(2). Let A′ = ⊕_{l}A′_{l}.

The L-function of the motive Sym^{2}h^{1}(E) is defined, for ℜ(s) > 2, by a Dirichlet series given by an Euler product

L(Sym^{2}E, s) = ∏_{r} P_{r}(r^{−s})^{−1}.

The product is over all primes r, and the polynomial P_{r}(X) := det(1 − Frob_{r}^{−1}X | V_{l}^{′I_{r}}), where I_{r} is an inertia subgroup at r, Frob_{r} is an arithmetic Frobenius element of Gal(\overline{ℚ}/ℚ) and l is any prime different from r. These Euler factors may be determined explicitly, see [CS]. Suffice it to say here that if r is a prime of good reduction of E then P_{r}(X) = (1 − α_{r}^{2}X)(1 − β_{r}^{2}X)(1 − rX), α_{r} and β_{r} being the eigenvalues of Frob_{r} on T_{l}.

Following [BK] (Section 3), for p ≠ l (including p = ∞) let

H_{f}^{1}(ℚ_{p}, V′_{l}) = ker(H^{1}(D_{p}, V′_{l}) → H^{1}(I_{p}, V′_{l})).

The subscript f stands for “finite part”. D_{p} is a decomposition subgroup at a prime above p, I_{p} is the inertia subgroup, and the cohomology is for continuous cocycles and coboundaries. For p = l let

H_{f}^{1}(ℚ_{l}, V′_{l}) = ker(H^{1}(D_{l}, V′_{l}) → H^{1}(D_{l}, V′_{l} ⊗ B_{cris}))

(see Section 1 of [BK] for definitions of Fontaine’s ring B_{cris}). Let H_{f}^{1}(ℚ, V′_{l}) be the subspace of elements of H^{1}(ℚ, V′_{l}) whose local restrictions lie in H_{f}^{1}(ℚ_{p}, V′_{l}) for all primes p. There is a natural exact sequence

0 ⟶ T′_{l} ⟶ V′_{l} ⟶^{π} A′_{l} ⟶ 0.

Let H_{f}^{1}(ℚ_{p}, A′_{l}) = π_{∗}H_{f}^{1}(ℚ_{p}, V′_{l}). Define the l-Selmer group H_{f}^{1}(ℚ, A′_{l}) to be the subgroup of elements of H^{1}(ℚ, A′_{l}) whose local restrictions lie in H_{f}^{1}(ℚ_{p}, A′_{l}) for all primes p. Note that the condition at p = ∞ is superfluous unless l = 2. In future, p will always be a finite prime. Define the Shafarevich-Tate group

⨿ ⨿ = ⊕_{l} \frac{H_{f}^{1}(ℚ, A′_{l})}{π_{∗}H_{f}^{1}(ℚ, V′_{l})}.

3. The Bloch-Kato formula, fudge factors

Let f(z) = ∑_{n=1}^{∞} a_{n}q^{n} (q = e^{2πiz}) be the normalised (a_{1} = 1) newform of weight 2 and level N associated with the elliptic curve E. (For p ∤ N, the number of points on E/𝔽_{p} is 1 + p − a_{p}.) Let ϕ : X_{0}(N) → E be a modular parametrisation. Let c be the associated Manin constant, i.e. ϕ^{∗}ω = c · 2πif(z)dz, where ω is a Néron differential on E.

The symmetric square L-function L(Sym^{2}E, s) is closely related to the Rankin convolution ∑_{n=1}^{∞} a_{n}^{2}n^{−s}, and L(Sym^{2}E, 2) may be evaluated using the Rankin-Selberg method [R]. Careful comparison of this with the conjectural formula of Bloch and Kato [BK] leads to formula (10) of [Fl1]:

(1)  \frac{deg ϕ}{Nc^{2}} ∏_{r∈S^{±}} \frac{r}{r ± 1} = \frac{# ⨿⨿}{#H^{0}(ℚ, A′)#H^{0}(ℚ, A′(−1))} ∏_{r≤∞} c_{r}.

Here, S^{±} are certain sets of primes of bad additive reduction. In the examples we look at later, E is always semistable, so S^{±} are empty. For precise definitions see [Fl1]. Likewise, the c_{r} are certain “fudge factors”. The following corrected Lemma 1 of [Fl1] provides us with what we need to know about them. If j is the j-invariant of the elliptic curve E and r is a finite prime, let d_{r} = −ord_{r}(j). Lemma 3.1. c_{r} = 1 for all but finitely many r. Up to powers of 2 and 3, and powers of r if r is a prime of bad reduction, we have

c_{r} = \begin{cases}1 & if d_{r} ≤ 0 or r = ∞; \\ #E(ℚ_{r})[d_{r}] & if d_{r} > 0. \end{cases}

The proof is essentially identical to the proof of Lemma 1 in [Fl1]. To apply the lemma, we need to be able to calculate #E(ℚ_{r})[d_{r}] in the case d_{r} > 0. In the proof of Lemma 1 of [Fl1], the l-part of E(ℚ_{r})[d_{r}] is isomorphic to ℤ_{l}(1)/d_{r} ⊕ ℤ_{l}/d_{r} as an I_{r}-module, but not necessarily as a Gal(\overline{ℚ}_{r}/ℚ_{r})-module, so the formula given there is not always correct.

We need to use the following proposition, due to Tate. A published proof may be found in [Si2] (see Lemma 5.1, Theorem 5.3 and Corollary 5.4). Proposition 3.2. Let E be an elliptic curve defined over ℚ_{r}, with ord_{r}(j) =: −d_{r} < 0. There is a unique q ∈ ℚ_{r} such that j(E) = \frac{1}{q} + 744 + 196884q^{2} + . . ..

(1) If E has split multiplicative reduction then there is an isomorphism of

groups:

E(ℚ_{r}) ≃ ℚ_{r}^{∗}/q^{ℤ}.

(2) If E has either non-split multiplicative reduction or additive reduction then there exists a quadratic extension L of ℚ_{r} such that

E(ℚ_{r}) ≃ {u ∈ L^{∗}/q^{ℤ} : N_{L/ℚ_{r}}(u) ∈ q^{ℤ}/q^{2ℤ}}.

With little trouble one may deduce the following lemma, which tells us the l-part of c_{r} in certain cases. Lemma 3.3. As above, suppose that E/ℚ_{r} with d_{r} > 0. Let l ≠ r be an odd prime with l^{a} exactly dividing d_{r}.

(1) If E has split multiplicative reduction then #E(ℚ_{r})[l^{a}] = l^{b+c}, where l^{b} = gcd(l^{a}, r − 1) and c = max{e ≤ a : (q/r^{d_{r}}) is a l^{e}-power (mod r)}.

(2) If E does not have split multiplicative reduction then

#E(ℚ_{r})[l^{a}] = gcd(l^{a}, r + 1).

(For example, l^{b} is the number of ℚ_{r}-rational roots of unity of order dividing l^{a}.) Note that j is the product of 1/q and a 1-unit in ℚ_{r}. Since 1-units in ℚ_{r} are l^{e}-powers, “q/r^{d_{r}}” may be replaced by “jr^{d_{r}}” in the above lemma.

Cremona’s table of Hecke eigenvalues includes the eigenvalues of the Atkin-Lehner involutions W_{p} for p | N. If p is a prime of multiplicative reduction, that reduction is split or non-split according as the eigenvalue of W_{p} is −1 or +1 respectively.

In the examples we look at later, l is a prime of good reduction, so certainly l ≠ r when d_{r} > 0. Also, since l ∤ N, the factor N in (1) has trivial l-part. By Corollary 4.2 of [M], if N is squarefree and E is a strong Weil curve within its isogeny class (as in all the examples we examine later), then the Manin constant c is at worst a power of 2, and has trivial l-part for odd prime l.

4. Global torsion

Next we look at the factors appearing in the denominator of (1). Lemma 4.1. If l is an odd prime and E[l] is an irreducible representation of Gal(\overline{ℚ}/ℚ) over 𝔽_{l} then #H^{0}(ℚ, A′) and #H^{0}(ℚ, A′(−1)) both have trivial l-part.

Proof. Via the Weil pairing, E[l] is dual to E[l](−1), as a Galois module. Hence

E[l] ⊗ E[l](−1) ≃ Hom_{𝔽_{l}}(E[l], E[l])

as modules for 𝔽_{l}[Gal(\overline{ℚ}/ℚ)]. Symmetric tensors correspond to linear maps of zero trace. H^{0}(ℚ, E[l] ⊗ E[l](−1)) corresponds to Hom_{𝔽_{l}[Gal(\overline{ℚ}/ℚ)]}(E[l], E[l]), which by Schur’s Lemma consists of scalar multiples of the identity. The only such map having zero trace is the zero map. Hence the l-part of #H^{0}(ℚ, A′(−1)) is trivial. Since E[l] is not isomorphic to E[l](1) as a Galois module, #H^{0}(ℚ, A′) also has trivial l-part. ■

The following comes from Proposition 21 of [Se], and provides us with a practical way of applying the previous lemma. Proposition 4.2. Suppose that N is squarefree (i.e. E is semistable). If l > 3 is a prime then E[l] is irreducible unless a_{p} ≡ 1 + p (mod l) for all primes p ∤ lN.

If E is semistable and a_{p} ≡ 1 + p (mod l) for all primes p ∤ lN then the composition factors of E[l] are 𝔽_{l} and 𝔽_{l}(1) (by Proposition 21 of [Se]). Using E[l] ⊗ E[l](−1) ≃ Hom_{𝔽_{l}}(E[l], E[l]), it is easy to prove the following lemma. Lemma 4.3. Let l be an odd prime, E/ℚ an elliptic curve.

(1) If E[l] ≃ 𝔽_{l} ⊕ 𝔽_{l}(1) then l | #H^{0}(ℚ, A′(−1)) and l | #H^{0}(ℚ, A′).

(2) If E[l] has a submodule but not a quotient isomorphic to 𝔽_{l} (i.e. if E has a rational point of order l, but is not l-isogenous to an elliptic curve with a rational point of order l) then l | #H^{0}(ℚ, A′_{l}) but the l-part of #H^{0}(ℚ, A′(−1)) is trivial.

(3) If E[l] has a submodule but not a quotient isomorphic to 𝔽_{l}(1) (i.e. if E has no rational point of order l, but is l-isogenous to an elliptic curve with a rational point of order l) then the l-parts of both #H^{0}(ℚ, A′) and #H^{0}(ℚ, A′(−1)) are trivial.

Lemma 4.4. Let l be an odd prime, E/ℚ an elliptic curve with primes r and r′ of multiplicative reduction such that l ∤ ord_{r}(j) and l^{2} ∤ ord_{r′}(j). Suppose either that l ∤ (r′ − 1) (split case) or that l ∤ (r′ + 1) (non-split case). Then the divisibilities in the above lemma are exact.

Proof. We prove part (2) to show the idea. There is an obvious Gal(\overline{ℚ}/ℚ)- equivariant map from E[l] to E[l](1) which maps the quotient of E[l] isomorphic to 𝔽_{l}(1) to the submodule of E[l](1) isomorphic to 𝔽_{l}(1), and this spans #H^{0}(ℚ, A′[l]). We need to show it is not divisible by l. Suppose for a contradiction that θ :

E[l^{2}] → E[l^{2}](1) is a Gal(\overline{ℚ}/ℚ)-equivariant map dividing it by l. Choose a ℤ/l^{2}ℤbasis \{v_{1}, v_{2}\} for E[l^{2}] such that under the Tate parametrisation of E(\overline{ℚ}_{r}), v_{1} corresponds to a primitive l^{2}-root of unity, and v_{2} corresponds to a chosen q^{1/l^{2}}, q being the Tate parameter of E.

If E has a rational point of order l then by (2) of Lemma 3.3, it is actually impossible for E to have non-split multiplicative reduction at r′, unless l | (r′ + 1). Hence, under our hypotheses, E has split multiplicative reduction at r′, and l ∤ (r′ − 1).

The extension of ℚ_{r} obtained by adjoining a primitive l^{2}-root of unity is unramified, and the further extension obtained by adjoining q^{1/l^{2}} is Kummer of degree l^{2} (because l ∤ ord_{r}(j)). Hence we can choose an element σ ∈ Gal(\overline{ℚ}/ℚ) (in an inertia subgroup at r) such that with respect to the basis {v_{1}, v_{2}}, the action of σ is repre-

sented by the matrix (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}). Suppose that θ is represented by a matrix (\begin{matrix}a & b \\ c & d \end{matrix}).

The matrices for σ and θ must commute. (Note that (ℤ/l^{2}ℤ)(1) is unramified at r, so twisting makes no difference to the action of σ.) It follows that, in ℤ/l^{2}ℤ, c = 0 and a = d. For zero trace, we must have a = d = 0. The reduction (mod l) of this matrix represents our element of H^{0}(ℚ, A′[l]). To avoid this reduction being trivial, we must have l ∤ b.

The image of θ is now a Gal(\overline{ℚ}/ℚ)-stable, rank one ℤ/l^{2}ℤ-submodule of E[l^{2}], generated by a point P of exact order l^{2}. Let χ : Gal(\overline{ℚ}/ℚ) → (ℤ/l^{2}ℤ)^{∗} be the character by which Gal(\overline{ℚ}/ℚ) acts on this submodule. Since lP is a rational point, χ(τ) ≡ 1 (mod l) for all τ ∈ Gal(\overline{ℚ}/ℚ), and χ is a character of order l.

Since l ∤ (r′ −1), the rational point lP cannot be represented by a root of unity in the Tate parametrisation of E(ℚ_{r′}). Necessarily l | ord_{r′}(j), but since l^{2} ∤ ord_{r′}(j), the character χ must be ramified at r′. But an abelian extension of ℚ of prime degree l ≠ r′, ramified at r′, must be a subfield of the cyclotomic field ℚ(ζ_{r′}), which is impossible since l ∤ (r′ − 1). ■

5. The construction of elements in a Selmer group

Theorem 5.1. Let l be an odd prime, E/ℚ an elliptic curve with primes r and r′ of multiplicative reduction (the latter split) such that l ∤ ord_{r}(j), l ∤ (r′ − 1) and l^{2} ∤ ord_{r′}(j). Suppose that E has a rational point P of infinite order and a rational point Q of order l. Suppose also that E has good reduction at l, and that for any prime p of bad reduction, E(ℚ_{p}) has no elements of order l^{2}. Then the Selmer group H_{f}^{1}(ℚ, A′_{l}) contains an element of order l.

Proof. There is a natural injection ψ from E(ℚ)/lE(ℚ) into H^{1}(ℚ, E[l]). Let c be the image of the point P of infinite order. Since E(ℚ) contains the point Q of order l, E[l] has a submodule isomorphic to 𝔽_{l}, with quotient 𝔽_{l}(1). Hence A′[l] := Sym^{2}(E[l]) has a submodule isomorphic to E[l], with quotient 𝔽_{l}(2). Since H^{0}(ℚ, 𝔽_{l}(2)) is trivial, we get an injection θ from H^{1}(ℚ, E[l]) to H^{1}(ℚ, A′[l]). Let c′ = θ(c) = θ(ψ(P)). Given the assumptions we have made, as in the proof of Lemma 4.4, H^{0}(ℚ, A′_{l}) = H^{0}(ℚ, A′[l]). It follows that the image in H^{1}(ℚ, A′[l]) of H^{0}(ℚ, A′_{l})/lH^{0}(ℚ, A′_{l}) is generated by θ(ψ(Q)), hence that the image d′ of c′ in H^{1}(ℚ, A′_{l}) is non-zero. We need to show that, for every finite prime p, res_{p}(d′) ∈ H_{f}^{1}(ℚ_{p}, A′_{l}).

Since l is a prime of good reduction, one may prove the local condition at p = l using Fontaine-Lafaille modules, as in the proof of Proposition 9.2 of [Du1].

Next consider a prime p ≠ l of good reduction. As is well-known (Proposition 2.1 in Chapter 8 of [Si1]), the class c ∈ H^{1}(ℚ, E[l]) is unramified at p. Hence the class d′ ∈ H^{1}(ℚ, A′_{l}) is unramified at p. Since A′_{l} is unramified at p (a prime of good reduction), H_{f}^{1}(ℚ_{p}, A′_{l}) is equal to (not just contained in) the kernel of the map from H^{1}(ℚ_{p}, A′_{l}) to H^{1}(I_{p}, A′_{l}), where I_{p} is an inertia subgroup at p (see line 3 of p. 125 of [Fl3]). Hence d′ ∈ H_{f}^{1}(ℚ_{p}, A′_{l}).

Finally, suppose that p ≠ l is a prime of bad reduction. It is easy to check (using Tate curves) that H_{f}^{1}(ℚ_{p}, V′_{l}) = {0}, so we need to show that res_{p}(d′) = 0. Let E_{1}(ℚ_{p}) be the kernel of reduction (mod p). Then E_{1}(ℚ_{p})/lE_{1}(ℚ_{p}) is trivial, and E(ℚ_{p})/E_{1}(ℚ_{p}) is finite, so the class of P in E(ℚ_{p})/lE(ℚ_{p}) may be represented by some l-power torsion point R ∈ E(ℚ_{p}). (What we have really done here is just to confirm that the image of E(ℚ_{p}) in H^{1}(ℚ_{p}, A_{l}) is H_{f}^{1}(ℚ_{p}, A_{l}) = {0}.) By assumption, R must be a point of order dividing l. Then res_{p}(c′) comes from the class in H^{0}(ℚ_{p}, A′_{l})/lH^{0}(ℚ_{p}, A′_{l}) represented by the image of R ⊗ Q + Q ⊗ R. This implies that res_{p}(d′) ∈ H^{1}(ℚ_{p}, A′_{l}) is zero, as required. ■

6. Examples–Rank zero with rational point of order 5

We use Cremona’s tables to examine the first 14 semistable strong Weil curves (ordered by conductor N) with 5 ∤ N, rank zero and a rational point of order l = 5. We use his labelling for the curves. There would be too many to go all the way to N = 7000. “deg” is the 5-part of the degree of the modular parametrisation. Using Lemmas 3.1 and 3.3, we compute C, defined to be the 5-part of the product ∏ c_{r} of local fudge factors, and using Lemmas 4.3 and 4.4 we compute (or bound) D, defined to be the 5-part of #H^{0}(ℚ, A′)#H^{0}(ℚ, A′(−1)). The minimal discriminant is Δ and “#5-⨿ ⨿?” is the order of the 5-part of ⨿ ⨿ predicted by the Bloch-Kato formula (1). Note that since any bad reduction is multiplicative, d_{r}, which was defined to be −ord_{r}(j), is also ord_{r}(Δ).

Name  Δ  C D deg #5-⨿ ⨿?
11A1  −11^{5}  5^{2} ≥ 5^{2} 1  ≥ 1
38B1  −2^{5}19  5 5  1  1
57C1  −3^{10}19 5 5  1  1
58B1  −2^{10}29 5 5  1  1
66C1 2^{10}3^{5}11 5^{2} 5  5  1
118B1 −2^{10}59 5 5  1  1
158C1  2^{20}79  5 5  1  1
186B1 −2^{5}3^{5}31 5^{2} 5  5  1
203A1  −7^{5}29  5 5  5^{0}  1
246B1 −2^{25}3^{5}41 5^{3} 5  5^{2}  1
286D1 −2^{5}11^{2}13^{5} 5^{2} 5  5  1
366B1 −2^{5}3^{5}61 5^{2} 5  5  1
426A1 −2^{5}3^{5}71 5^{2} 5  5  1
537E1 −3^{10}179 5 5  1  1

There is no particular reason to expect elements of order 5 in ⨿ ⨿, and in each case the predicted order of the 5-part of ⨿ ⨿ is 1. The presence of rational 5-torsion forces #E(ℚ_{r})[5] to be divisible by 5. This produces powers of 5 dividing those c_{r} such that 5 | d_{r}. These are beautifully balanced by powers of 5 dividing deg(ϕ). See especially the example 246B1.

7. Examples–No rational point of order 5, modular parametrisation degree not divisible by 5

This seems to apply to most curves, and is not a very interesting case. A random selection of ten is 14A1, 26A1, 38A1, 57B1, 66A1, 69A1, 82A1, 102C1, 122A1 and 138B1. For each of these there is no bad r such that 5 | d_{r}, hence C = 1. Also, in each example there is no congruence a_{p} ≡ 1 + p (mod 5) for all p ∤ 5N (this may be checked using Cremona’s table of Hecke eigenvalues), so by Lemmas 4.2 and 4.1 we find that D = 1. Hence, the 5-part of # ⨿⨿ is predicted to be trivial in all these examples.

8. Examples–5 dividing modular parametrisation degree

Here are the first nine for which 5 | deg(ϕ), followed by the first four for which 5^{2} | deg(ϕ). As before, we look only at semistable, strong Weil curves with 5 ∤ N.

Name  Δ  #E(ℚ)_{tors.} C D deg #5-⨿ ⨿?
46A1 −2^{10}23  2  1 1 5  5
66C1 2^{10}3^{5}11  10  5^{2} 5 5  1
67A1  −67  1  1 1 5  5
77B1 −7^{6}11^{3}  3  1 1 5  5
78A1 −2^{16}3^{5}13  2  1 1 5  5
89B1  −89^{2}  2  1 1 5  5
106D1 −2^{5}53  1  1 1 5  5
114B1 2^{2}3^{5}19  2  1 1 5  5
114C1 2^{20}3^{3}19  4  5 1 5  1
483A1 −3^{5}723^{3}  1  5 1 5^{2}  5
503C1 −503  1  1 1 5^{2}  5^{2}
506D1 2^{5}11^{5}23  1  5 1 5^{2}  5
573B1 3^{5}191  1  5 1 5^{2}  5

In the example 506D1, which has split multiplicative reduction at 11, we find that 11^{5}j = −2^{12}31^{3} is not a 5^{th} power (mod 11), so that #E(ℚ_{11})[5] is only 5, not 5^{2} (recall Lemmas 3.1 and 3.3). Also for 506D1, the reduction at 2 is non-split, so #E(ℚ_{2})[5] is only 1, not 5. In several of the other examples there are primes r of non-split multiplicative reduction such that 5 | d_{r}.

In each of the above examples, since 5 | deg(ϕ), according to (1) either C or #⨿⨿ has to be divisible by 5, and it seems that it is sometimes one, sometimes the other (and sometimes both). In several cases 5 divides some d_{r} without there being any rational point of order 5, though not always with the result that 5 divides C, since the reduction at r may be non-split.

Note also that Proposition 2 of [Fl1] states that if l > 3 is a prime of good reduction such that the natural map from Gal(\overline{ℚ}/ℚ) to Aut(E[l]) is surjective, and if l | d_{r}, (for some bad r such that d_{r} > 0) then l | deg(ϕ). He argues that since l | d_{r}, the representation of Gal(\overline{ℚ}/ℚ) on E[l] is unramified at r. By work of Ribet, the modular form f is then congruent (mod l) to some other modular form, of level dividing N/r. Hence l is a “congruence prime” and divides deg(ϕ). The condition about the image of Gal(\overline{ℚ}/ℚ) is satisfied for semistable curves not satisfying a_{p} ≡ 1 + p (mod l) for all p ∤ lN, by Proposition 21 of [Se]. In particular, it is satisfied in all the above examples except 66C1. Of course, the main theorem of [DFG1] applies to all these examples (except 66C1).

9. Examples–Rank one with rational point of order 5

We examine all semistable strong Weil curves of conductor N ≤ 7000, with 5 ∤ N, rank at least one and a rational point of order 5. An asterisk signifies rank two, otherwise the rank is one.

Name  Δ  C D deg #5-⨿ ⨿?
123A1  −3^{5}41  5 5  5  5
302A1  −2^{15}151  5 5  5  5
834G1 −2^{10}3^{5}139 5^{2} 5 5^{2}  5
874E1  2^{5}1923^{5}  5^{2} 5 5^{2}  5
1147B1  31^{2}37^{5}  5 5  5  5
1293E1  3^{15}431  5 5  5  5
1479F 1 −3^{5}17^{5}29^{2} 5^{2} 5 5^{2}  5
1526E1 −2^{5}7^{5}109 5^{2} 5 5^{2}  5
1717C1  17^{5}101  5 5  5  5
2651C1  −11^{5}241 5^{2} ≥ 5 5^{2}  ≥ 5
2786D1 −2^{5}7^{5}199 5^{2} 5 5^{2}  5
2869B1 19^{5}151 5 5 5 5
3026D1 −2^{5}17^{5}89 5^{2} 5 5^{2}  5
3206E1 2^{10}7^{10}2294 5^{2} 5 5^{2}  5
4043A1 −13^{5}311 5 5 5 5
4774J1 −2^{15}7^{5}11^{2}31 5^{2} 5 5^{3}  5^{2}
4854C1 −2^{15}3^{10}809 5^{2} 5 5^{2}  5
5034E1 −2^{15}3^{5}839 5^{2} 5 5^{3}  5^{2}
5074D1 −2^{10}43^{5}59 5^{2} 5 5^{2}  5
∗5302I1 2^{5}11^{5}241 5^{3} 5 5^{3}  5
6782E1  2^{30}3391  5 5 5^{2}  5

In stark contrast to the examples in Section 6 (rank zero with a rational point of order 5), here the predicted order of ⨿ ⨿ is always divisible by 5. This is in keeping with Theorem 5.1, which always produces a candidate for an element of order 5 in ⨿ ⨿, though in 5 out of the 21 examples the technical conditions of the theorem are not satisfied. For 2651C1 there is no prime r′ of multiplicative reduction such that 5 ∤ (r′ − 1). For each of 302A1, 1717C1, 2786D1 and 2869B1, there is a prime r (151, 101, 199 and 151 respectively) of bad reduction such that E(ℚ_{r}) has a point of order 25. The example 5302I1 is puzzling. Since E(ℚ) has rank 2, our construction should give us two independent elements of order 5 in ⨿ ⨿, yet the predicted order of the 5-part of ⨿ ⨿ is only 5.

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University of Sheffield, Department of Pure Mathematics, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, U.K.

   E-mail address: n.p.dummigan@shef.ac.uk