Notes on Kato-Siegel functions

A special function.

First consider the elliptic curve E_{τ} over ℂ corresponding to the lattice ℤ+τℤ, where Im(τ) > 0.

Consider the function

Θ(u, τ) = q^{\frac{1}{12}} (t^{\frac{1}{2}} − t^{− \frac{1}{2}} ) ∏_{n>0}(1 − q^{n}t)(1 − q^{n}t^{−1}),

where q = e^{2πiτ} and t = e^{2πiu}. For any integer D prime to 6, construct the function

θ_{D}^{E_{τ} /ℂ} := (−1)^{\frac{D−1}{2}} Θ(u, τ)^{D^{2}}Θ(Du, τ)^{−1}.

Then θ_{D}^{E_{τ} /ℂ} enjoys the following properties.

(i) θ_{D}^{E_{τ} /ℂ} has divisor D^{2}(e) − ker[×D]. (ii) For any isogeny α : E → E′ of degree prime to D between two such curves, α_{∗}θ_{D}^{E/ℂ} = θ_{D}^{E′/ℂ}. (iv) • θ_{−D} = θ_{D} (This is obvious.)

• θ_{1} = 1

• [×M]_{∗}θ_{MC} = θ_{C}^{M^{2}} ∈ O^{∗}(E − ker[×C]) (In particular, [×D]_{∗}θ_{D} = 1.)

• θ_{C} ∘ [×M] = θ_{MC}/θ_{M}^{C^{2}} ∈ O^{∗}(E − ker[×MC]) (The reason for the numbering will be become apparent in a moment.)

Aim.

We wish to exhibit the algebraic nature of this phenomenon, and show that it generalizes to elliptic curves over arbitrary schemes, behaving well in families. Let f : E → S be an elliptic curve over an arbitrary scheme S. Let ω_{E/S} be the invertible sheaf

f_{∗}Ω_{E/S}^{1} = e^{∗}Ω_{E/S}^{1},

which, since Ω_{E/S}^{1} is free along the fibres of f, is

= x^{∗}Ω_{E/S}^{1}

for any section x ∈ E(S).

We want to assign to each E/S, and D prime to 6, a section

θ_{D}^{E/S}

of O^{∗}(E − ker[×D]) such that

(i) θ_{D}^{E/S} (as a rational function) has divisor D^{2}(e) − ker[×D]. (ii) The assignment is compatible with isogeny. Precisely, for any isogeny α : E → E′ of degree prime to D between elliptic curves over S, we have α_{∗}θ_{D}^{E/S} = θ_{D}^{E′/S}.

(iii) The assignment is compatible with base change. Precisely, for any base change

\xymatrix{E \ar[d]_{f} & E ×_{S} S′ \ar[d] \ar[l]_{g} \\ S & S′ \ar[l]}

we have g^{∗}(θ_{D}^{E/S}) = θ_{D}^{E×_{S}S′/S′}. (iv) • θ_{−D} = θ_{D}

• θ_{1} = 1

• [×M]_{∗}θ_{MC} = θ_{C}^{M^{2}} ∈ O^{∗}(E − ker[×C]) (In particular, [×D]_{∗}θ_{D} = 1.)

• θ_{C} ∘ [×M] = θ_{MC}/θ_{M}^{C^{2}} ∈ O^{∗}(E − ker[×MC])

(v) In the case where E is an elliptic curve over ℂ corresponding to the lattice ℤ + τℤ, Im(τ) > 0, the section θ_{D}^{E/ℂ} is as before. That is,

θ_{D}^{E/ℂ} = (−1)^{\frac{D−1}{2}} Θ(u, τ)^{D^{2}}Θ(Du, τ)^{−1}.

Note that (i) and (ii) determine θ_{D}^{E/S} uniquely. (By (i) any other possible assignment is uθ_{D}^{E/S}, for some u ∈ O^{∗}(S). Applying (ii) with α = [×2] and [×3] (2 and 3 are prime to D) gives

uθ_{D}^{E/S}
=α_{∗}(uθ_{D}^{E/S})
=α_{∗}(u)α_{∗}θ_{D}^{E/S}
=α_{∗}(u)θ_{D}^{E/S}
= both u^{4}θ_{D}^{E/S} and u^{9}θ_{D}^{E/S},

whence u^{4} = u = u^{9}, ⇒ u = 1.

Example. Suppose S = Spec k, k algebraically closed, α : E → E′ separable. Then (ii) becomes

∏_{x∈E(k)} ; α(x)=yθ_{D}^{E/k}(x) = θ_{D}^{E′/k}(y) ∀y ∈ E′(k),

which is the familiar distribution relation.

Proof of main result. First note that if S = Spec k, then ker[×D] − D^{2}(e) is principal. (Because D is odd, the sum of ker[×D] on the elliptic curve is 0.)

• To give a rule θ_{D} satisfying (i) and (iii) is equivalent to giving an isomorphism of line bundles

O_{E}(ker[×D]) → O_{E}(D^{2}e)

compatible with base change. Taking any fibre of E/S, we have the situation S = Spec k above; hence the line bundles are indeed isomorphic when restricted to any fibre. Then, our task is equivalent to finding, for each E/S, a trivialization of

e^{∗}O_{E}(ker[×D]) ⊗ e^{∗}O_{E}(D^{2}e)^{∨},

which is

=e^{∗}O_{E}(ker[×D]) ⊗ e^{∗}O_{E}(−D^{2}e)
=e^{∗}[×D]^{∗}O_{E}(e) ⊗ e^{∗}O_{E}(−D^{2}e)
=e^{∗}O_{E}(e)^{⊗(1−D^{2})}
=ω_{E/S}^{⊗(D^{2}−1)},

compatible with base change.

Now we know the set of nowhere-vanishing sections of ω^{⊗12d} over the moduli stack – that is, the collections, of a section of ω_{E/S}^{⊗12d} for each E/S, compatible with base change and isogeny – is {±Δ^{d}}, for any d ∈ ℤ, where Δ is the discriminant. Then (since (D, 6) = 1 ⇒ D ≡ 1 mod 12) we have 2 nonvanishing sections of ω^{⊗(D^{2}−1)}, that is, ±Δ(E/S)^{(D^{2}−1)/12}. Let ±ϕ_{D}^{E/S} be the corresponding functions on E −ker[×D]. So both E/S ↦ ±ϕ_{D}^{E/S} satisfy (i) and (ii).

Change base so that α factors as a product of isogenies of prime degree. Thus to verify (ii) we may assume deg α = p prime.

The quotient g_{p}(E/S, α) := α_{∗}ϕ^{E/S}(ϕ^{E′/S})^{−1} ∈ O^{∗}(S) is compatible with base change. The modular stack M_{Γ_{0}(N)} classifies pairs (E/S, α) where α : E → E′ is a cyclic isogeny of degree N. So g_{p} is a modular unit ∈ Γ(M_{Γ_{0}(p)}, O^{∗}), and so g_{p}(E/S, α) = ±1 ∀(E/S, α). And the sign depends only on p.

• To determine the sign evaluate g_{p}(E/𝔽_{p}, Fr_{E}). Now Fr_{E∗} : κ(E)^{∗} → κ(E)^{∗} is the norm map. So for p odd, g_{p}(E, Fr_{E}) = 1. In particular this does not depend on our choice of ±ϕ_{p}^{E/S}. For p = 2, though, replacing one by the other replaces g_{2} by −g_{2}. Therefore exactly one of ±ϕ^{E/S} will make θ_{2} satisfy (ii).

• We check (iv). θ_{−D} also satisfies (i) and (ii), which uniquely determine θ_{D}. So θ_{−D} = θ_{D}. The constant 1 satisfies (i) and (ii) for D = 1; hence θ_{1} = 1.

[×M]_{∗}θ_{MC} has divisor M^{2}(C^{2}(e) − ker[×C]) and is compatible with base change, so [×M]_{∗}θ_{MC} = ϵθ_{C}^{M^{2}}, ϵ = ±1. Now (ii) gives

ϵθ_{C}^{M^{2}} =[×M]_{∗}θ_{MC}
=[×M]_{∗}[×2]_{∗}θ_{MC}
=[×2]_{∗}[×M]_{∗}θ_{MC}
=[×2]_{∗}(ϵθ_{C}^{M^{2}})
=ϵ^{4}θ_{C}^{M^{2}}
=θ_{C}^{M^{2}}

and so ϵ = 1.

Using D = M and C = 1 produces [×D]_{∗}θ_{D} = θ_{1}^{D^{2}} = 1.

Now θ_{C} ∘ [×M] and θ_{MC}/θ_{M}^{C^{2}} both have divisor

C^{2} ker[×M] − ker[×MC];

hence their ratio is a unit compatible with base change. So (i) and (ii) give the result as before.

• A final matter is to check that condition (v) holds – that is, that our θ_{D}^{E/S} is indeed a generalization of the analytic θ_{D}^{E_{τ} /ℂ} given at first. This found by calculating that F (u, τ) := Θ(u, τ)^{D^{2}}Θ(Du, τ)^{−1} is a function on E_{τ} (≈ ℂ/(ℤ+ τℤ)) with divisor D^{2}(e)−ker[×D], and which is sl_{2}(ℤ)-invariant. Then F (u, τ) is a constant multiple of θ_{D}^{E/S} for E = E_{τ}, the constant being independent of τ.

Consideringt a curve E_{τ} defined over ℝ with two real connected components (for example Y^{2} = X^{3} − X, where τ = i) we calculate that

[×2]_{∗}F (u, τ) = (−1)^{\frac{D−1}{2}} F (u, τ).

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