Second Example

Our second example will be the rational elliptic curveE of conductor 144 given by

E :Y2 =X3−1.

The corresponding newformφ is given by

φ(z) = q+ 4q7+ 2q13−8q19−5q25+ 4q31−10q37−8q43+ 9q49+O(q50).

Here Mφ = 144. Since (H1) and (H2) are satisfied, there exists a N such

that S3/2(N, χ, φ)6={0}, where 144|(N/2) and again χ2 =χtriv. We assume thatχ is the trivial character. Using Corollary 3.6.4 for computing Shimura’s decomposition, we find that at the level 576, the space S3/2(576, χ, φ) 6={0}; and this space has a basis {f1, f2, f3, f4} where f1, f2, f3 and f4 have the

following q-expansion: f1(z) =q−q25+ 5q49−6q73−6q97+O(q100) := ∞ X n=1 anqn f2(z) =q5+q29−q53−2q77+O(q100) := ∞ X n=1 bnqn f3(z) =q13−2q61+q85+O(q100) := ∞ X n=1 cnqn f4(z) =q17−q41−q89+O(q100) := ∞ X n=1 dnqn.

Doing similar calculations as in the previous example it turns out that

f

Nφ= 576. Using Waldspurger’s Theorem there exists a functionAφon square-

free numbers such thatS3/2(576, χ, φ) = U(576, φ, Aφ). Following the compu-

tations we get that U(576, φ, Aφ) is spanned by P ∞

n=1Aφ(n

sc_)n1/4Q

pcp(n)q n

where the choices for c2 include the characteristic functions of 1, 5 modulo

Q×2 2

, while the choices for c3 are characteristic functions of 1, 2 moduloQ×3 2

. The following lemma is a special case of a standard theorem on the torsion of Mordell elliptic curves (i.e. elliptic curves of the formY2 _=X3_+B). For the proof see [8, page 52].

Lemma 4.5.8. Let E be as above and let n be a square-free integer. Then

En,tor ∼=Z/2Z unless n=−1 in which case E−1,tor ∼=Z/6Z.

The discriminant of the model E−1 : Y2 = X3 + 1 is −432 = 24 ×33 which is sixth-power free. By Lemma 4.4.3, Ω(E−n) = Ω(E−1)/

√

n.

We have the following lemma on root numbers which can be proved on similar lines as Proposition 4.5.4.

Lemma 4.5.9. Let E be as above. For n positive square-free the following holds. (i) If ν3(n) = 0 then, W(E−n/Q) =          1 n ≡1,5 (mod 8) −1 n ≡3,7 (mod 8) −1 n even.

(ii) If ν3(n) = 1 then, W(E−n/Q) =          1 n/3≡1,5 (mod 8) −1 n/3≡3,7 (mod 8) 1 n even.

Finally, we have the following theorem.

Theorem 4.5.10. Let E :Y2 =X3−1. Let

f =f1/2 +f2+ √ 2f3+ √ 3f4 := ∞ X n=1 enqn.

Let n6= 1 5 be positive square-free integer such that n≡1,2 (mod 3). Then,

L(E−n,1) = ΩE−1 √ n ·e 2 n. (4.7)

Further assuming BSD, if E−n has rank zero then, |_X(E−n/Q)|= Q4

pcp ·e2_n

where the Tamagawa numbers c2 = 3 if n ≡ 1 (mod 8), c2 = 1 if n ≡ 3,5,7 (mod 8); c3 = 2; cp = #E−1(Fp)[2] for p | n, p 6= 3; and cp = 1 for all other primes p.

Proof. It is to be noted that using Theorem 3.8.10, we can prove that an is

non-zero only for n ≡ 1 (mod 24), bn is non-zero only for n ≡ 5 (mod 24),

cn is non-zero only for n ≡ 13 (mod 24) and dn is non-zero only for n ≡ 17

(mod 24). Thus we can choosef as in the theorem (the choice for coefficients of fi in f are done using similar calculations as in Theorem 4.5.5). Using

Lemma 4.5.9, we see that both sides of equation (4.7) vanish if n ≡ 1,2 (mod 3) and n ≡ 3,7 (mod 8). So it is enough to consider the other cases. Recall that Aφ(n)2 = L(E−n,1) by Theorem 4.3.4. The proof of the first

statement now follows.

5_{In the case n= 1 we still have L(E}

−n,1) =

Ω_√E−1 n ·e

2

n, but since |E−1,tor| = 6 we get

that|_X(E−n/Q)|= Q36

pcp·e

2

For the second statement, we use Lemma 4.5.8 and substitute Ω(E−n) =

Ω(E−1)/

√

n in the equation (4.5). The calculation for Tamagawa numbers cp

are done as before (see Corollary 4.5.6).

In order to consider the case of E−n when 3 | n we try to look at the

space S3/2(1728, χtriv, φ) but it turns out that this space is equal to the space

S3/2(576, χtriv, φ). Hence we do not get any new information.

Another possible way to deal with this situation is to work with the quadratic character χ3 = 3_·

, instead of the trivial character. Our algorithm shows that S3/2(576, χ3, φ) = {0} and S3/2(1728, χ3, φ) has a basis consisting of g1, g2, g3 and g4 where gi’s are as follows:

g1 =q3−q75+5q147−6q219−6q291+O(q300), g2 =q39−2q183+q255+O(q300),

g3 =q15+q87−q159−2q231+O(q300), g4 =q51−q123−q267+O(q300). Waldspurger’s Theorem now asserts the existence of a functionAφ(which now

depends on χ3 and φ) on Nsc such that Aφ(n)2 = L(E−3n,1). Note that gi’s

have non-zero n-th coefficient only for n ≡ 3, 6 (mod 9). Further if n = 3m

then L(E−3n,1) = L(E−m,1). This leads us to obtain exactly the same results

as in Theorem 4.5.10.

Remark. It is to be noted that we cannot apply Waldspurger’s Theorem to the elliptic curveE0 given by

E0 :Y2 =X3+ 1

since it is easy to check that the hypothesis (H1) is not satisfied. However,

E0 = E−1, hence by Theorem 4.5.10 we get information about the positive

n-th quadratic twists of E0 forn with 3 _-n. Further note thatE3 is isogenous toE−1, hence L(En,1) = L(E−3n,1) for alln. Thus computation of L(E−3n,1)

for n positive square-free will lead to a formula for L(En,1) and hence for