ON THE L-FUNCTION OF THE CURVES y^{2} = x^{5} + A

MICHAEL STOLL AND TONGHAI YANG

Abstract. Let C_{A}/ℚ be the curve y^{2} = x^{5} + A, and let L(s, J_{A}) denote the L-series of its Jacobian. Under the assumption that the sign in the functional equation for L(s, J_{A}) is +1, we evaluate the critical value L(1, J_{A}) in terms of the value of a theta series for ℚ(\sqrt{5}) depending on A at a CM point coming from ℚ(ζ_{5}).

1. Introduction

In this note, we study the L-function of the genus two curve

(1.1)  C_{A} : y^{2} = x^{5} + A

and its arithmetic implication to the curve and its Jacobian J_{A}. Here A is a (non-zero) rational number. The arithmetic of C_{A} has been studied by the first author in [St1, St2] in a more general setting. Although elliptic curves have been extensively studied and a lot of progress has been made, little is known for higher genus curves or higher dimensional abelian varieties. The curves C_{A} are interesting, since their Jacobians J_{A} are not modular in the usual sense in that they are not isogenous to any quotient of J_{0}(N), the Jacobian of a modular curve, for any integer N. (This is because J_{A} does not have real multiplication over ℚ.) To put it another way, there is no classical (normalized) eigenform f of weight 2 such that

L(s, J_{A}) = ∏_{σ:ℚ(f)⟶ℂ}L(s, f^{σ})

where ℚ(f) is the subfield of ℂ generated by the Fourier coefficients of f. So well-known results on the Birch and Swinnerton-Dyer conjecture (as for example Kolyvagin and Logachev [KL]) do not apply directly.

However, since J_{A} has complex multiplication by ℤ[ζ_{5}] over E = ℚ(ζ_{5}), there is a Hecke character η_{A} of E such that

(1.2)  L(s, J_{A}) = L(s, η_{A}) ,

Date: February 2, 2003.

1991 Mathematics Subject Classification. 11G40, 11G30, 11G15, 11F27, 11F41.

T.Y. was partially supported by NSF grant DMS-0070476.

compare [St2]. By results of Jacquet-Langlands [JL, Prop. 12.1], this implies that there is a Hilbert modular form h_{A} over F = ℚ(\sqrt{5}) such that

L(s, J_{A}) = L(s, h_{A}) .

One then easily obtains that

L(s, J_{A}/F) = L(s, h_{A})L(s, h_{A}^{τ})

where τ is the non-trivial automorphism of F/ℚ; therefore J_{A} is modular in the usual sense over F.

In this note, we will give a method to express the central L-value L(1, η_{A}) in terms of the value of a certain theta series for F, which depends on A, at a certain CM point coming from E (which may also depend on A). We carry this out explicitly in the special case A = 1, where the result is as follows (see Theorem 4.1).

L(1, J_{1}) = L(1, η_{1}) = \frac{π^{2} |I|^{2}}{5^{3/4}(1 − cos(4π/5) − sin(2π/5))}

with

I = ∑_{a,b∈ℤ,a+b odd}exp(−\frac{2π}{5} (sin(\frac{2π}{5})^{3}(a + b\sqrt{5})^{2} + sin(\frac{4π}{5})^{3}(a − b\sqrt{5})^{2})) .

In particular, L(1, J_{1}) ≠ 0. A 2-descent procedure as described in [St3] shows that J_{1}(ℚ) is finite (of order 10). So the rank part of the Birch and Swinnerton-Dyer conjecture is verified in this case. See Section 6 at the end of this note for more comments on the Birch and Swinnerton-Dyer conjecture for the Jacobians J_{A}.

We mention that a similar explicit formula has been obtained in [RVY, Ya2, Ya3] for CM number fields E such that E has no roots of unity other than {±1} and that every prime of E above 2 is split in E/F. So the work presented here can be viewed as a complement. The main new ingredient is to work out what happens above 2 in the case considered here, where tenth roots of unity are present in E and 2 is inert, see the remark after Proposition 3.1 and Section 5.

This note is organized as follows. In Section 2, we introduce notations and collect necessary information about the character η_{A}. In Section 3, we use [Ya1] to obtain a general formula for the central L-value L(1, η_{A}). In Section 4, we obtain the explicit formula for L(1, η_{1}) alluded to above, leaving a local technical computation to Section 5.

We would like to thank Y. Tian, Don Zagier, S.-W. Zhang and the anonymous referee for useful discussions, comments and suggestions.

2. Preliminaries

Let E = ℚ(ζ_{5}) and F = ℚ(\sqrt{5}). We denote by O = O_{F} the ring of integers of F. We fix an embedding of E into ℂ by identifying ζ = ζ_{5} with e^{2πi/5}. Let Nα denote the norm of α ∈ E from E to ℚ. Set λ = 1 − ζ and denote by ρ the character of O_{E,λ}^{×} of conductor λ^{2} that is given by ρ|_{ℤ_{5}^{×}} = 1 and ρ(ζ) = ζ^{−1}. (This is χ_{λ^{2}} from [St2].)

Fix δ = ζ^{−2} − ζ^{2} ∈ E. Then Δ = δ^{2} ∈ F. We also fix a CM type Φ of E as Φ = {σ_{2}, σ_{4}}, where σ_{a}(ζ) = ζ^{a} = e^{2πia/5}. Note that σ_{2}(δ) = 2i sin(2π/5) and σ_{4}(δ) = 2i sin(4π/5). In particular,

(2.1)  |σ_{2}(δ)σ_{4}(δ)| = \sqrt{5} .

In the following, (\frac{·}{·}) denotes the usual Legendre symbol, whereas (\frac{·}{·})_{n} denotes the nth power residue symbol on E. We denote by 𝔸 = 𝔸_{F} the adele ring and by 𝔸^{×} the idele group of F. Let χ_{A} = η_{A}| · |_{𝔸}^{1/2} be the unitary counterpart of η_{A}. Then

L(\frac{1}{2}, χ_{A}) = L(1, η_{A}) .

Lemma 2.1. The restriction of χ_{A} to 𝔸^{×} coincides with the quadratic Hecke character of F associated to E/F.

Proof: By [St2, Cor. 3.2 and Lemma 3.6], the Größencharakter corresponding to χ_{A} is given (on elements α prime to 10A) by

χ̃_{A}(α) = (\frac{A}{α})_{2} (\frac{4A}{α})_{5}^{4} (\frac{α}{λ})_{2} ρ(α)σ_{2}(α)σ_{4}(α)(Nα)^{−1/2} .

If α is in F, then the first two power residue symbols take the value 1. The third symbol gives the quadratic Größencharakter belonging to E/F, ρ is trivial, and the last three factors cancel. □

We will fix a ‘canonical’ additive character ψ = ∏ ψ_{v} of 𝔸/F as follows:

ψ_{v}(x) = e^{−2πiλ_{v}(x)} ,

where

λ_{v} : F_{v} ⟶^{Tr} ℚ_{p} ⟶ ℚ_{p}/ℤ_{p} ↪ ℚ/ℤ ,

if v is finite, and λ_{v}(x) = −x if v is real. Let ψ_{E} = ψ ∘ Tr_{E/F}. Note that the λ_{v} used here is the negative of that used in [La].

We need some information on χ_{A}. We will suppose that A is an integer not divisible by 5 or the tenth power of some prime number. Translating the results of [St2, Sect. 3] into the language used here, we obtain the following.

Proposition 2.2. Let w be some place of E that lies over the place v of F. The ‘root number’ denotes Tate’s local root number ϵ(\frac{1}{2}, χ_{A,w}, ψ_{E_{w}}(\frac{1}{2} ·)).

(1) If w is infinite (and complex), then χ_{A,w}(z) = |z|/z. The root number is i, and χ_{A,w}(δ) = −i.

(2) If w is finite and inert in E/F, and w does not divide 10A, then χ_{A,w}(π) = −1 when π ∈ F_{v} is a uniformizer, and χ_{A,w} is unramified. The root number and χ_{A,w}(δ) are both 1.

(3) If w ≠ 2 is finite and inert in E/F, and w divides A, then χ_{A,w}(π) = −1 when π ∈ F_{v} is a uniformizer. The conductor exponent of χ_{A,w} is 1, and on O_{E,w}^{×}, we have

χ_{A,w}(α) = (\frac{α}{π})_{2}^{e} (\frac{α}{π})_{5}^{e} ,

where e is the valuation of A at w (or at the prime p below w). The root number is (−1)^{e+1} and χ_{A,w}(δ) = (−1)^{e} (\frac{−1}{N_{F/ℚ}π})^{e}.

(4) If w = 2, then χ_{A,2}(2) = −1. The conductor exponent f_{2} of χ_{A,2} is determined as follows. Write A = 2^{e}B with B odd. Then

f_{2} = \begin{cases}0 & if e = 8 and B ≡ 1 mod 4, \\ 1 & if e < 8, e is even and B ≡ 1 mod 4, \\ 2 & if e is even and B ≡ −1 mod 4, \\ 3 & if e is odd. \end{cases}

On O_{E,2}^{×}, we have

χ_{A,2}(α) = (\frac{2}{Nα})^{e} (\frac{−1}{Nα})^{(B−1)/2} (\frac{α}{2})_{5}^{e+2} .

The root number is (−1)^{1+f_{2}+e} and χ_{A,2}(δ) = (−1)^{e}.

(5) If w = λ, let q(A) = (A^{4} − 1)/5. Then the conductor exponent f_{λ} of χ_{A,λ} is 1 if 5|q(A) and 2 otherwise. On O_{E,λ}^{×}, we have

χ_{A,λ}(α) = (\frac{α}{λ})_{2} ρ(α)^{q(A)} .

Furthermore, χ_{A,λ}(δ) = − (\frac{A}{5}), and the root number is −1 if 5|q(A) and (\frac{Aq(A)}{5}) otherwise.

(6) Let N_{A} = 2^{f_{2}} ∏_{2≠p|A} p. Then the global root number of χ_{A} is given by

− (\frac{AN_{A}}{5} ) if 5|q(A) and by (\frac{q(A)N_{A}}{5} ) otherwise.

We will assume throughout this paper that the global root number of χ_{A} is 1. Then there is an α ∈ F^{×} (unique up to norm from E^{×}) such that for all places v of F,

∏_{w|v} ϵ(\frac{1}{2}, χ_{A,w}, ψ_{E_{w}}(\frac{1}{2} ·))χ_{A,w}(δ) = ϵ_{v}(α) ,

where the product is over places w of E above v, ϵ(\frac{1}{2}, χ_{A,w}, ψ_{E_{w}}(\frac{1}{2} ·)) are the local root numbers as in the preceding proposition, and ϵ_{v} is the local part of the Hecke character belonging to E/F. (This is automatically satisfied when v is split in E/F.) The following lemma follows easily from the preceding proposition.

Lemma 2.3. Suppose the global root number of χ_{A} is 1. Then we can take

α = 2^{1+f_{2}} ∏_{p} p ,

where the product is over primes p ≠ 2 such that p | A, but leaving out the primes p ≡ −1 mod 20 with v_{p}(A) odd. In particular, if A is a square such that v_{2}(A) < 8 then we can take

α = ∏_{2≠p|A} p .

Proof: The first statement follows from the proposition at all places v ≠ \sqrt{5}. For this last place, the claim then follows from the product formula and the fact that the global root number is 1. The second statement follows from the first, since f_{2} = 1 and all v_{p}(A) are even. □

3. A general formula

We will use the notation in [Ya1]. Let S(𝔸) be the space of Schwartz functions on the adeles 𝔸 of F. Let G = E^{1} be the norm one subgroup of E^{×}, and view it as a unitary group of one variable. Associated to the data (χ_{A}, ψ, δ, α) chosen above, there is a Weil representation ω = ω_{α,χ_{A}} of G(𝔸) on S(𝔸). For each character η of G(𝔸), there is an associated automorphic representation of G(𝔸) given by

{θ_{ϕ}(η)(g) : ϕ ∈ S(𝔸), g ∈ G(𝔸)}

Here [G] = G(F)\G(𝔸) and

θ_{ϕ}(η)(g) = ∫_{[G]} ∑_{x∈F} ω_{α,χ_{A}}(gh)ϕ(x) dh .

The representation is either η or zero. Whether it is η or not is related to the nonvanishing of the central L-value L(\frac{1}{2}, χ_{A}η̃) where η̃(z) = η(z/z̄). Here η has nothing do with η_{A} and will always be the trivial character 1 in this note. What we need is the following. Let ϕ = ∏_{v} ϕ_{v} ∈ S(𝔸) such that ϕ_{v} is the same as ϕη̄_{v} (with η_{v} trivial) in [Ya1, Thm. 2.15] for v ∤ 2. We will choose ϕ_{2} later on and let it

6 MICHAEL STOLL AND TONGHAI YANG

be any locally constant function on F_{2} with compact support for now. Then the proof of [Ya1, Thms 1.11 and 2.15] gives the following formula.

C_{1}C_{2} \frac{L(1, η_{A})}{L(1, ϵ)} = 2 |θ_{ϕ}(1)(1)|^{2} .

Here

C_{1} = ∏_{2≠v|A, inert}(1 + q_{v}^{−1})^{−1} ∏_{v|A, split} q_{v}^{−1}(1 − q_{v})^{−2}

is the constant in [Ya1, Thm. 2.15], and

(3.1)  C_{2} = ∫_{G_{2}}⟨ω_{α,χ_{A},2}(g)ϕ_{2}, ϕ_{2}⟩dg ,

where G_{2} = G(E_{2}) = E_{2}^{1}. Moreover, L(1, ϵ) = π^{2}/25.

Set G_{2,1} = {g ∈ G_{2} : g ≡ 1 mod 2}. Then G_{2} = ⟨ζ⟩ × G_{2,1}. Set

U = G_{2,1} ∏_{v≠2, nonsplit} G_{v} ∏_{v split} O_{v}^{×};

then G_{𝔸} = G(F)U and G(F) ∩ U = {±1}. So

θ_{ϕ}(1)(1) = ∫_{[G]} ∑_{x∈F} ω_{α,χ_{A}}(g)ϕ(x)dg

= \frac{1}{2} ∫_{U} ∑_{x∈F} ω_{α,χ_{A}}(g)ϕ(x)dg

= \frac{1}{2} ∑_{x∈F} ∏_{v} I_{v}(x) .

Here

I_{v}(x) = \begin{cases}∫_{O_{v}^{×}} ω_{α,χ_{A},v}(g)ϕ_{v}(x)dg & if v is split, \\ ∫_{G_{v}} ω_{α,χ_{A},v}(g)ϕ_{v}(x)dg & if v ≠ 2 is nonsplit, \\ ∫_{G_{2,1}} ω_{α,χ_{A},2}(g)ϕ_{2}(x)dg & if v = 2. \end{cases}

By definition of ϕ_{v}, it is easy to see that I_{v}(x) = ϕ_{v}(x) when v ∤ 2A. So we have proved Proposition 3.1. Let the notation be as above. Then

L(1, η_{A}) = \frac{π^{2}}{50C_{1}C_{2}} |∑_{x∈F} ∏_{v∤2A}ϕ_{v}(x) ∏_{v|2A} I_{v}(x)|^{2}

when C_{2} is nonzero.

There is always a choice of ϕ_{2} making C_{2} nonzero, e.g., we can take ϕ_{2} to be a unitary eigenfunction of (G_{2}, ω_{α,χ_{A},2}); then C_{2} = 1, compare [Ya1]. However, this eigenfunction is not explicitly constructed in the case p = 2. That is why the primes above 2 are assumed to be split in E/F in [RVY] and [Ya2, Ya3] as mentioned in the introduction. The fact that E has extra roots of unity also causes some technical problems. In Section 5, we will exhibit an explicit ϕ_{2} that is almost an eigenfunction of (G_{2}, ω_{α,χ_{A},2}) and makes C_{2} ≠ 0. We refer to Section 5 for details and to Corollary 5.8 for the result.

Let char(X) denote the characteristic function of the set X.

Lemma 3.2.

ϕ_{v}(x) = \begin{cases}char(O_{v})(x) & if v ∤ 10 A ∞ and α ∈ O_{v}^{×}, \\ |2σ_{j}(αδ^{3})|^{1/4}e^{−π|σ_{j}(αδ^{3})|σ_{j}(x)^{2}} & if v = σ_{j} ∈ \{σ_{2}, σ_{4}\}. \end{cases}

Proof: The split case follows from [Ya1, Thm. 2.15]. The nonsplit case with 10Aα ∈ O_{v}^{×} follows from [Ya2, Prop. 1.2 and Cor. 1.4]. Finally, the infinite case follows from [Ya2, Lemma 1.1]. □

Note that |σ_{j}(αδ^{3})| = −i σ_{j}(−αδ^{3}), since α is totally positive and σ_{j}(δ) is purely imaginary with positive imaginary part. Hence

e^{−π|σ_{j}(αδ^{3})|σ_{j}(x)^{2}} = e^{πiσ_{j}(−αδ^{3})σ_{j}(x)^{2}},

and the expression under the absolute value in Prop. 3.1 is a theta function evaluated at the CM point given by −αδ^{3}.

4. The case A = 1

We now specialize to the case A = 1. By Lemma 2.3, we can then take any α ∈ N_{E/F} E^{×}. We choose α = 1/5 (which is a square in F). Then we have

ϕ\sqrt{5}(x) = 5^{1/4} char(O\sqrt{5})(x) .

This follows from [Ya2, Prop. 1.2 and Cor. 1.4] again, since χ_{A,λ} has conductor (exponent) 1.

For v = 2, we will prove in Cor. 5.8 below that we can take ϕ_{2} = char(\frac{1}{2} + O_{2}). The constant C_{2} then has the value

C_{2} = \frac{1 − cos(\frac{4π}{5}) − sin(\frac{2π}{5})}{5} ,

and I_{2}(x) = ϕ_{2}(x). Note that here e = 0 and (\frac{−1}{α}) = 1.

Now, putting everything together (and using (2.1)), we finally get an explicit expression.

Theorem 4.1. We have

L(1, η_{1}) = C |I|^{2} > 0

with

C = \frac{π^{2}}{5^{3/4}(1 − cos(4π/5) − sin(2π/5))}

and

I = ∑_{x∈ \frac{1}{2} +O} exp(−\frac{π}{5} ((2 sin(\frac{2π}{5}))^{3}σ_{2}(x)^{2} + (2 sin(\frac{4π}{5}))^{3}σ_{4}(x)^{2})) .

5. Local computations at v = 2

In this section, we find a suitable ϕ_{2} and compute related terms in the case that f_{2} = 1 in the notation of Prop. 2.2.

We let A be arbitrary for now. To lighten notation, we drop the subscript 2. So F = ℚ_{2}(\sqrt{5}) is the unramified extension of ℚ_{2} of degree 2 and E = F (δ) = ℚ_{2}(ζ) is the unramified extension of ℚ_{2} of degree 4. We also have G = E_{2}^{1}. Let O = O_{F} = ℤ_{2}[\frac{1+\sqrt{5}}{2}] be the ring of integers of F. Set for k ≥ 1

Γ_{k} = {g ∈ G : g ≡ 1 mod 2^{k}} , Γ′_{k} = {g = x + yδ ∈ G : y ∈ 2^{k}O} .

We also write simply χ instead of χ_{A,2}.

The following two lemmas on the structure of G are easily verified.

Lemma 5.1. The map a ↦ g_{a} = \frac{a+δ}{a−δ} is a bijection between ℙ^{1}(F) = F ∪ \{∞\} and G. Moreover

(1) g_{a} ∈ Γ_{1} if and only if a^{2} ≢ Δ mod 4.

(2) g_{a} ∈ Γ_{1} − Γ′_{2} if and only if a ∈ O^{×} and a^{2} ≢ Δ mod 4.

(3) For k > 1, g_{a} ∈ Γ′_{k} − Γ_{k} if and only if a ∈ 2^{k−1}O.

(4) For k > 1, g_{a} ∈ Γ_{k} if and only if 1/a ∈ 2^{k−1}O.

Lemma 5.2.

(1) G = ⟨ζ⟩ × Γ_{1}.

(2) We have

ζ^{±1} = g_{∓ \frac{1+\sqrt{5}}{2}} = \frac{−1+\sqrt{5}}{4} ∓ \frac{1+\sqrt{5}}{4}δ ,
ζ^{±2} = g_{∓ \frac{3−\sqrt{5}}{2}} = −\frac{1+\sqrt{5}}{4} ∓ \frac{1}{2}δ .

(3) The group Γ_{1} is generated by Γ′_{3}, g_{2}, g_{1+\sqrt{5}} and g_{\frac{1−\sqrt{5}}{2}} .

Recall that ([Ku, Prop. 4.8], see also [Ya1, (A1)])

ω_{α,χ}(g)ϕ = μ_{α,χ}(g)r(ι_{α}(g))ϕ ,

where r is Rao’s standard section [Rao],

μ_{α,χ}(g) = χ(δ(g − 1))γ_{F} (αy(1 − x)ψ)(Δ, −2y(1 − x))_{F}
= χ(−δy(g − 1))γ_{F} (αy(1 − x)ψ) ,

(note that 2(1 − x) is the norm of g − 1, so (Δ, 2(1 − x))_{F} = 1 and (Δ, −y)_{F} = ϵ(−y) = χ(−y)) and

ι_{α} : G ⟶ Sp(1) = SL_{2}(F) , g = x + yδ ↦ (\begin{matrix}x & Δ^{2}αy \\ \frac{y}{Δα} & x \end{matrix}) .

Let w = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}). Write

(5.1) ι_{α}(g)w = (\begin{matrix}x^{−1} & Δ^{2}αy \\ 0 & x \end{matrix}) w (\begin{matrix}1 & −\frac{y}{Δαx} \\ 0 & 1 \end{matrix}) = m(x^{−1})n(Δ^{2}αxy)wn(−\frac{y}{Δαx})

for g ∈ Γ_{1}, and

(5.2)  ι_{α}(g) = (\begin{matrix}\frac{Δα}{y} & x \\ 0 & \frac{y}{Δα} \end{matrix}) w (\begin{matrix}1 & \frac{Δαx}{y} \\ 0 & 1 \end{matrix}) = m(\frac{Δα}{y})n(\frac{xy}{Δα})wn(\frac{Δαx}{y})

for g ∈ G − Γ_{1}. Here

m(a) = (\begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix}) , n(b) = (\begin{matrix}1 & b \\ 0 & 1 \end{matrix}) .

Set

ξ_{α,χ}(g) = \begin{cases}μ_{α,χ}(g)γ_{F} (2Δαxyψ) & if g = x + yδ ∈ Γ_{1}, \\ μ_{α,χ}(g) & if g ∈ G − Γ_{1}. \end{cases}

Then one has by (5.1) and [Rao, Thm. 3.5, Cor. 4.3]

(5.3)  ω_{α,χ}(g)ϕ(u) = ξ_{α,χ}(g)r(ι_{α}(g)w)ϕ̂(−u)

for every ϕ ∈ S(F) and g ∈ Γ_{1}, where

ϕ̂(u) = r(w)ϕ(u) = ∫_{F} ϕ(v)ψ(−uv) dv

is the Fourier transform of ϕ.

From now on, we will assume that f_{2} = 1 in the notation of Prop. 2.2. Then we can take α ∈ O^{×} to be a unit. We have χ(2) = −1, and on O_{E}^{×}, χ is a character of order 5 which is trivial on 1 + 2O_{E}. Furthermore, χ(δ) = 1 and χ is trivial on O^{×}.

Lemma 5.3. Assume α ∈ O^{×}. For β ∈ \frac{1}{2}O, set f_{β} = char(β + O). Then

ω_{α,χ}(g)f_{β} = ξ_{α,χ}(g)ψ(\frac{Δ^{2}αy}{2x} β^{2})f_{β}

for all g = x + yδ ∈ Γ_{1}.

Proof: We have the following explicit formulas (cf. [Rao, Thm. 3.5]).

r(n(b))ϕ(u) = ψ(\frac{1}{2}bu^{2})ϕ(u)
r(m(a))ϕ(u) = |a|^{1/2}ϕ(au)
r(w)ϕ(u) = ϕ̂(u)
r(m(a)n(b)wm(a′)n(b′)) = r(m(a))r(n(b))r(w)r(m(a′))r(n(b′))

We first have to compute f̂_{β}.

f̂_{β}(u) = ∫_{F} f_{β}(v)ψ(−uv) dv
= ∫_{O} ψ(−u(β + v)) dv
= ψ(−uβ) char(O)(u)

(use that { \atop char(O)} = char(O)). Now we use (5.1) and (5.3) together with above explicit formulas and the fact that x ∈ 1 + 2O and y ∈ 2O when g ∈ Γ_{1}. This gives

ω_{α,χ}(g)f_{β}(u) = ξ_{α,χ}(g)r(ι_{α}(g)w)r(−w)f_{β}(u)
= ξ_{α,χ}(g)r(m(x^{−1})n(Δ^{2}αxy)wn(−\frac{y}{Δαx}))ψ(βu) char(O)(u)
= ξ_{α,χ}(g)r(m(x^{−1})n(Δ^{2}αxy)w)ψ(−\frac{y}{2Δαx}u^{2})ψ(βu) char(O)(u)

(use that −\frac{y}{2Δαx}u^{2} ∈ O)

= ξ_{α,χ}(g)r(m(x^{−1})n(Δ^{2}αxy)w)ψ(βu) char(O)(u)
= ξ_{α,χ}(g)r(m(x^{−1})n(Δ^{2}αxy)) ∫_{O} ψ(βv)ψ(−uv) dv
= ξ_{α,χ}(g)r(m(x^{−1})n(Δ^{2}αxy)) char(O)(u − β)
= ξ_{α,χ}(g)r(m(x^{−1})n(Δ^{2}αxy)) char(β + O)(u)
= ξ_{α,χ}(g)r(m(x^{−1}))ψ(\frac{1}{2}Δ^{2}αxyu^{2}) char(β + O)(u)
= ξ_{α,χ}(g)|x|^{−1/2}ψ(\frac{Δ^{2}αy}{2x}u^{2}) char(β + O)(x^{−1}u)

(use that x ∈ 1 + 2O and β ∈ \frac{1}{2}O)

= ξ_{α,χ}(g)ψ(\frac{Δ^{2}αy}{2x}u^{2}) char(β + O)(u)

(use that u ∈ β + O and that \frac{Δ^{2}αy}{2x} ∈ O)

= ξ_{α,χ}(g)ψ(\frac{Δ^{2}αy}{2x}β^{2})f_{β}(u) .
□

Lemma 5.4. Let α ∈ O^{×} and let φ(g) = ξ_{α,χ}(g). Then φ is a character on Γ_{1} that does not depend on α. It is trivial on the subgroup generated by Γ′_{3}, g_{1+\sqrt{5}} and g_{\frac{1−\sqrt{5}}{2}} , and φ(g_{2}) = −1.

Proof: Write g_{a} = x + yδ, then

x = \frac{a^{2} + Δ}{a^{2} − Δ} , y = \frac{2a}{a^{2} − Δ} , 1 − x = −\frac{2Δ}{a^{2} − Δ} .

We will drop the subscript F on γ and the Hilbert norm residue symbol. We have

γ(αy(1 − x)ψ)γ(2Δαxyψ) = γ(2Δx(1 − x), ψ)γ(ψ)^{2}(αy(1 − x), 2Δαxy)

= γ(−(a^{2} + Δ), ψ)γ(−1, ψ)(−αaΔ, αaΔ(a^{2} + Δ))

= γ(a^{2} + Δ, ψ)(αaΔ, a^{2} + Δ) .

Here we have used the fact that γ(ψ)^{2} = γ(−1, ψ)^{−1} = −1 = γ(−1, ψ). Now, for g_{a} ∈ Γ_{1} and α ∈ O^{×}, we have (α, a^{2} + Δ) = 1 (since a^{2} + Δ is a square times something that is a square mod 4). So

φ(g) = ξ_{α,χ}(g)(α, a^{2} + Δ) = χ(−δy(g − 1))(aΔ, a^{2} + Δ)γ(a^{2} + Δ, ψ)

is independent of α. In particular, we may assume α = 1. Taking β = 0 in the last lemma, we see that ξ_{1,χ} is a character of Γ_{1} (since ω_{1,χ} is a true representation).

Since Γ_{1} is a pro-2 group and χ on O_{E}^{×} has order five, χ(−δy(g − 1)) = (−1)^{ord_{2}(y)+ord_{2}(g−1)}.

Recall that (a, b) with b ∈ O^{×} only depends on b mod 2^{3}, and when also a ∈ O^{×}, it only depends on b mod 2^{2}.

For g ∈ Γ_{3}, one has 1/a ∈ 4O, so ord_{2}(y) = 1 − ord_{2}(a) = ord_{2}(g − 1), and a^{2} + Δ = a^{2}(1 + Δ/a^{2}) is a square, hence φ(g) = 1.

For g ∈ Γ′_{3} − Γ_{3}, one has a ∈ 4O, so ord_{2}(y) = 1 + ord_{2}(a) and ord_{2}(g − 1) = 1. Furthermore, a^{2} + Δ = Δ(1 + a^{2}/Δ) is Δ times a square. Since γ(Δ, ψ) = 1, and (a, Δ) = (−1)^{ord_{2}(a)}, we again get φ(g) = 1.

For a = 1 + \sqrt{5}, one has that a^{2} + Δ is a square mod 4 but not mod 8 and ord_{2}(a) = 1, so (aΔ, a^{2} + Δ) = −1 and γ(a^{2} + Δ, ψ) = γ(Δ, ψ) = 1. Since ord_{2}(y) = 2, ord_{2}(g − 1) = 1, we have φ(g_{a}) = 1.

For a = \frac{1−\sqrt{5}}{2}, one has a^{2} + Δ = −1 and aΔ = −\sqrt{5}, so (aΔ, a^{2} + Δ) = (−\sqrt{5}, −1) = −1 and γ(a^{2} + Δ, ψ) = γ(−1, ψ) = −1. Since ord_{2}(y) = 1 = ord_{2}(g − 1), we have φ(g_{a}) = 1.

Finally, for a = 2, one has a^{2} +Δ = (\frac{−1+\sqrt{5}}{2})^{2} and ord_{2}(y) = 2, ord_{2}(g −1) = 1, hence φ(g_{2}) = −1. □

Corollary 5.5. If α ∈ ℤ_{2}^{×}, then f_{\frac{1}{2}} is an eigenfunction of Γ_{1} with trivial character.

Proof: By Lemma 5.3, it suffices to show that

ψ(\frac{Δ^{2}αy}{8x} ) = ψ(\frac{αaΔ^{2}}{4(a^{2} + Δ)}) = \begin{cases}1 & if g ∈ Γ′_{3}, a = 1 + \sqrt{5}, or a = \frac{1−\sqrt{5}}{2}, \\ −1 & if a = 2. \end{cases}

This is trivially true for g ∈ Γ′_{3} (since then the argument of ψ is integral). The other three cases are easily verified. □

Lemma 5.6. For α ∈ O^{×} and g = ζ^{j} = x + yδ with 5 ∤ j, one has

r(ι_{α}(g))f_{\frac{1}{2}} (u) = \frac{1}{2}ψ(\frac{Δαx}{2y}(u^{2} − \frac{u}{x} + \frac{1}{4})) char(\frac{1}{2}O)(u) .

Proof: This is a computation similar to that in the proof of Lemma 5.3, but using (5.2) instead of (5.1). Note that x and y are both in \frac{1}{2}O^{×}. □

We need to compute ξ_{α,χ}(ζ^{j}).

Lemma 5.7. Let α ∈ ℤ_{2}^{×}. Then

ξ_{α,χ}(ζ^{j}) = −χ(ζ^{j} − 1) \begin{cases}1 & if j = ±1, \\ ± (\frac{−1}{α}) i & if j = ±2. \end{cases}

Furthermore, χ(ζ^{j} − 1) = ζ^{−(e+2)j}, where e = ord_{2}(A).

Proof: Since y ∈ F and ord_{2}(y) = −1, we have χ(−δy) = −1, which gives χ(−δy(ζ^{j} −1)) = −χ(ζ^{j} −1). The factor involving the Weil index can be computed explicitly (use that γ(aψ) only depends on a mod 4 when a is a unit).

Finally, we have

(\frac{ζ^{j} − 1}{2})_{5} ≡ (ζ^{j} − 1)^{3} ≡ 1 + ζ^{j} + ζ^{2j} + ζ^{3j} ≡ ζ^{−j} mod 2

and

χ(ζ^{j} − 1) = (−1)^{e} (\frac{ζ^{j} − 1}{2})_{5}^{e+2} = (\frac{ζ^{j} − 1}{2})_{5}^{e+2}

by Prop. 2.2, (4) and the fact that e is even (since f_{2} = 1). □

ON THE L-FUNCTION OF THE CURVES y^{2} = x^{5} + A 13

Corollary 5.8. Let α ∈ ℤ_{2}^{×} and denote e = ord_{2}(A). If we choose ϕ_{2} = f_{\frac{1}{2}} , then the constant C_{2} in Prop. 3.1 has the value

C_{2} = \frac{1}{5}(1 − cos((e + 2)\frac{2π}{5} ) + (\frac{−1}{α}) sin(2(e + 2)\frac{2π}{5} )) ,

and I_{2}(x) = ϕ_{2}(x).

Note that C_{2} ≠ 0 since e ∈ {0, 2, 4, 6} when f_{2} = 1. Proof: By (3.1), we have to compute

C_{2} = ∫_{G}⟨ω_{α,χ}(g)f_{\frac{1}{2}} , f_{\frac{1}{2}} ⟩ dg .

By Cor. 5.5, ω_{α,χ}(g)f_{\frac{1}{2}} = f_{\frac{1}{2}} for g ∈ Γ_{1}, so I_{2} = ϕ_{2}, and by Lemma 5.2, (1), G = ⟨ζ⟩ × Γ_{1}. Hence

C_{2} = \frac{1}{5} ∑_{j=0}^{4}⟨ω_{α,χ}(ζ^{j})f_{\frac{1}{2}} , f_{\frac{1}{2}} ⟩

= \frac{1}{5}(1 + \frac{1}{2} ∑_{j=1}^{4} ξ_{α,χ}(ζ^{j})ψ(\frac{Δα(x_{j} − 1)}{4y_{j}}))

((by Lemma 5.6) where ζ^{j} = x_{j} + y_{j}δ)

= \frac{1}{5}(1 + \frac{1}{2}ξ_{α,χ}(ζ) − \frac{1}{2}ξ_{α,χ}(ζ^{2}) − \frac{1}{2}ξ_{α,χ}(ζ^{3}) + \frac{1}{2}ξ_{α,χ}(ζ^{4}))

= \frac{1}{5}(1 − \frac{1}{2}(ζ^{e+2} + ζ^{−(e+2)}) + (\frac{−1}{α}) \frac{1}{2i}(ζ^{2(e+2)} − ζ^{−2(e+2)}))

(by Lemma 5.7)

= \frac{1}{5}(1 − cos((e + 2)\frac{2π}{5} ) + (\frac{−1}{α}) sin(2(e + 2)\frac{2π}{5} )) .

□

6. The Birch and Swinnerton-Dyer Conjecture

Since J_{A} has complex multiplication by ℤ[ζ_{5}] defined over E = ℚ(ζ_{5}), there is a Hecke character η_{A} of E such that

L(s, J_{A}) = L(s, η_{A}) ,

compare [Sh1, Thm. 20.9]. See also [St2], where it is shown that η_{A} has conductor

ν_{A} = λ^{f_{λ}}2^{f_{2}} ∏_{2≠p|A} p

if A is not divisible by 5 or the tenth power of a prime number.

Recently, the Gross-Zagier formula [GZ] and the work of Kolyvagin and Logachev [Ko, KL] have been extended to totally real number fields by S.-W. Zhang [Zh1, Zh2] and Y. Tian [Ti] respectively under some technical conditions. If these conditions can be removed to cover more general cases, our result will imply finiteness of J_{1}(ℚ) and of � (ℚ, J_{1}). In lack of those powerful results, our result still verifies the rank part of the Birch and Swinnerton-Dyer conjecture for J_{1} since the finiteness of the Mordell-Weil group of J_{1} can be proved by a 2-descent argument as in [St3]. In fact, J_{1}(ℚ) has order 10 and is generated by the divisor class [(0, 1) − (−1, 0)]. The Birch and Swinnerton-Dyer conjecture would imply that � (ℚ, J_{1}) is actually trivial, see below. The 2-descent shows at least that its 2-part is trivial.

It is interesting to compare the L-series value L(1, J_{A}) with the value predicted by the Birch and Swinnerton-Dyer conjecture. One version of it says that L(1, J_{A}) ≠ 0 if and only if J_{A}(ℚ) is finite, and then

(6.1)  L(1, J_{A}) = \frac{Ω(J_{A}) ∏_{p} c_{p}(J_{A})#� (ℚ, J_{A})}{#J_{A}(ℚ)^{2}} .

Here Ω(J_{A}) is the ‘real period’ of J_{A} over ℚ and c_{p}(J_{A}) are the Tamagawa numbers. It is known by a general result of Blasius [Bl] that

\frac{L(1, J_{A})}{Ω(J_{A})}

is rational. Since C_{A} has a rational point, #� (ℚ, J_{A}) should be a square [PS]. So it is very interesting to see whether its predicted or “analytic” value for A = 1

#� (ℚ, J_{1})_{an} = \frac{L(1, J_{1})#J_{1}(ℚ)^{2}}{Ω(J_{1}) ∏_{p} c_{p}(J_{1})} = \frac{C |I|^{2}#J_{1}(ℚ)^{2}}{Ω(J_{1}) ∏_{p} c_{p}(J_{1})}

is indeed an integer square (C and I are given in Theorem 4.1). This involves relating the value of a theta function at a CM point to the square root of the period of J_{1} in a precise way. G. Shimura proved in [Sh2] that the ratio is an algebraic number, which is not enough for our purpose. Pinning down which number field the ratio lies in is an important and non-trivial problem, since the period is in general defined only up to a scalar and thus the square root is defined only up to square roots of scalars.

We remark that all the invariants in (6.1) are computable except the order of the Shafarevich-Tate group, see [F+]. If one carries this through for the curves C_{A}, with A an integer not divisible by 5, one obtains the following.

ON THE L-FUNCTION OF THE CURVES y^{2} = x^{5} + A 15

Define

ω(p, A) = p^{4⌊v_{p}(A)/10⌋} · \begin{cases}1 & if v_{p}(A) ≡ 0, 1, 2, 3 mod 10; \\ p & if v_{p}(A) ≡ 4, 5, 6 mod 10; \\ p^{2} & if v_{p}(A) ≡ 7, 8, 9 mod 10; \end{cases}

and

c(p, A) = \begin{cases}5 & if 10 ∤ v_{p}(A) and A ∈ (ℚ_{p}^{×})^{2}; \\ 2 & if v_{p}(A) ≡ 5 mod 10 and p ≡ ±2 mod 5; \\ 4 & if v_{p}(A) ≡ 5 mod 10 and p ≡ −1 mod 5; \\ 16 & if v_{p}(A) ≡ 5 mod 10, A ∈ (ℚ_{p}^{×})^{5} and p ≡ 1 mod 5; \\ 1 & else. \end{cases}

Then let

ω_{p}(A) = \begin{cases}ω(p, A) & if p is odd; \\ ω(2, A) & if p = 2 and v_{2}(A) is odd; \\ ω(2, 2A) & if p = 2, v_{2}(A) is even and 2^{−v_{2}(A)}A ≡ 3 mod 4; \\ ω(2, 4A) & if p = 2, v_{2}(A) is even and 2^{−v_{2}(A)}A ≡ 1 mod 4; \end{cases}

and similarly

c_{p}(A) = \begin{cases}c(p, A) & if p ≠ 2, 5; \\ 1 & if p = 5 and 5 ∤ q(A); \\ 2 & if p = 5 and 5 | q(A); \\ c(2, A) & if p = 2 and v_{2}(A) is odd; \\ c(2, 2A) & if p = 2, v_{2}(A) is even and 2^{−v_{2}(A)}A ≡ 3 mod 4; \\ c(2, 4A) & if p = 2, v_{2}(A) is even and 2^{−v_{2}(A)}A ≡ 1 mod 4. \end{cases}

Also let

Ω_{0}(A) = \frac{πΓ(\frac{1}{10})Γ(\frac{3}{10})}{5Γ(\frac{3}{5})Γ(\frac{4}{5})} · \begin{cases}A^{−2/5} & if A > 0; \\ (−A)^{−2/5}/\sqrt{5} & if A < 0. \end{cases}

Then we have

c_{p}(J_{A}) = c_{p}(A) and Ω(J_{A}) = Ω_{0}(A) ∏_{p} ω_{p}(A) .

Furthermore,

#J_{A}(ℚ)_{tors} = \begin{cases}10 & if A ∈ (ℚ^{×})^{10}; \\ 5 & if A ∈ (ℚ^{×})^{2} \ (ℚ^{×})^{10}; \\ 2 & if A ∈ (ℚ^{×})^{5} \ (ℚ^{×})^{10}; \\ 1 & else. \end{cases}

For example, if A = 1, the Birch and Swinnerton-Dyer conjecture claims that

L(1, J_{1}) = \frac{πΓ(\frac{1}{10})Γ(\frac{3}{10})}{50Γ(\frac{3}{5})Γ(\frac{4}{5})} #� (ℚ, J_{1})

or equivalently

C|I|^{2} = \frac{πΓ(\frac{1}{10})Γ(\frac{3}{10})}{50Γ(\frac{3}{5})Γ(\frac{4}{5})} #� (ℚ, J_{1}).

Here C and I are given in Theorem 4.1. By numerical calculations, this would imply that � (ℚ, J_{1}) is trivial—the value for #� (ℚ, J_{1}) comes out as 1.0 to many decimal digits. So the Birch and Swinnerton-Dyer conjecture in this case implies a relation between theta values and Gamma values.

References

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[Ti] Y. Tian: Columbia University PhD thesis, 2003.

[Zh1] Shou-Wu Zhang: Heights of Heegner points on Shimura curves, Ann. of Math. 153 (2001), 27–147.

[Zh2] Shou-Wu Zhang: Gross-Zagier formula for GL_{2}, preprint (2001).

School of Engineering and Science, International University Bremen, P.O.Box 75 05 61, 28725 Bremen, Germany

  E-mail address: m.stoll@iu-bremen.de

Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA.

  E-mail address: thyang@math.wisc.edu