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A simple zero of the Weierstrass sigma function on the square lattice

Example

Let $Λ = Z + i Z$ , with oriented basis $ω_{1} = 1$ , $ω_{2} = i$ , and let $ζ = ζ_{Λ}$ , $σ = σ_{Λ}$ be the Weierstrass zeta and sigma functions. Then $σ (0) = σ (1) = 0$ , $σ^{'} (0) = 1$ and $σ^{'} (1) = - \exp (η_{1} / 2) \neq 0$ where $η_{1} = 2 ζ (1 / 2)$ : both zeros of $σ$ at $0$ and at $1$ are simple. Moreover $ζ$ has residue $1$ at both points.

Facts & Assumptions

Given: The square lattice $Λ = Z + i Z = Z ω_{1} + Z ω_{2}$ with $ω_{1} = 1$ , $ω_{2} = i$ , its Weierstrass functions $ζ = ζ_{Λ}$ and $σ = σ_{Λ}$ (Weierstrass $ζ$ and $σ$ functions), and $η_{1} = 2 ζ (ω_{1} / 2) = 2 ζ (1 / 2)$ .

[F1]

$Λ = Z ω_{1} + Z ω_{2}$ is a full complex lattice with oriented basis $ω_{1} = 1$ , $ω_{2} = i$ , and $ζ, σ$ are its Weierstrass zeta and sigma functions (Complex lattice and quotient torus, Weierstrass $ζ$ and $σ$ functions).

[F2]

$ζ$ is meromorphic on $C$ , holomorphic exactly on $C ∖ Λ$ , odd, and at every lattice point $λ \in Λ$ has a simple pole with principal part $(z - λ)^{- 1}$ and residue $1$ , with no other poles. $σ$ is entire and odd, its zero set is exactly $Λ$ and every zero is simple, $σ^{'} (0) = 1$ , and $σ^{'} (z) / σ (z) = ζ (z)$ for $z \in C ∖ Λ$ . Moreover the quasi-period laws hold for all $z \in C$ with poles matched: $ζ (z + ω_{j}) = ζ (z) + η_{j}$ and $σ (z + ω_{j}) = - \exp (η_{j} (z + ω_{j} / 2)) σ (z)$ for $j = 1, 2$ , where $η_{j} = 2 ζ (ω_{j} / 2)$ (Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions).

[F4]

The complex exponential satisfies $\exp (0) = 1$ and $\exp (z + w) = \exp (z) \exp (w)$ ; consequently $\exp (z) \exp (- z) = 1$ and $\exp (z) \neq 0$ for every $z \in C$ (The complex exponential by its power series, $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential). Its defining series also proves continuity: for $∣ u ∣ \leq 1$ , $n! \geq 2^{n - 1}$ for $n \geq 1$ gives $∣ \exp (u) - 1 ∣ \leq ∣ u ∣ \sum_{n \geq 1} 1 / n! \leq 2 ∣ u ∣$ . The addition law then gives $∣ \exp (v + u) - \exp (v) ∣ \leq 2 ∣ \exp (v) ∣ ∣ u ∣$ , so $\exp$ is continuous at every $v$ .

Verification

1.1F1F2

(Values at $0$ .) Since $0 \in Λ$ , [F2] gives $σ (0) = 0$ and $σ^{'} (0) = 1$ , and the zero of $σ$ at $0$ is simple; also $ζ$ has at $0$ a simple pole with residue $1$ .

1.2F1F2

(Residues of $ζ$ .) Both $0$ and $1$ lie in $Λ = Z ω_{1} + Z ω_{2}$ ; by [F2] the only poles of $ζ$ are the lattice points and each is simple with residue $1$ , so $ζ$ has residue $1$ at $0$ and at $1$ .

2.1F1F2step 1.1

( $σ (1) = 0$ .) Put $z = 0$ in the sigma quasi-period law for $j = 1$ : $ω_{1} = 1$ and $η_{1} = 2 ζ (1 / 2)$ , so $σ (1) = - \exp (η_{1} \cdot \frac{1}{2}) σ (0) = - \exp (η_{1} / 2) \cdot 0 = 0$ .

3.1F2F4step 1.1step 2.1

( $σ^{'} (1) = - \exp (η_{1} / 2) \neq 0$ .) For $u \neq 0$ the quasi-period law and step 2.1 give [F2, step 2.1] $\frac{σ (1 + u) - σ (1)}{u} = - \exp ⁣ (η_{1} (u + 1 / 2)) \frac{σ (u) - σ (0)}{u} .$ As $u \to 0$ , the second factor tends to $σ^{'} (0) = 1$ by step 1.1, and the exponential tends to $\exp (η_{1} / 2)$ by the continuity derived in [F4]. Thus $σ^{'} (1) = - \exp (η_{1} / 2) \neq 0$ by [F4]; simplicity and the residue $1$ of $ζ$ at $1$ also follow directly from [F2].

4.1

(Assembly.) Steps 1.1, 2.1 and 3.1 give $σ (0) = σ (1) = 0$ with simple zeros, $σ^{'} (0) = 1$ and $σ^{'} (1) = - \exp (η_{1} / 2) \neq 0$ ; step 1.2 gives residue $1$ of $ζ$ at both points. This is the asserted statement. ∎

Remarks

The whole example is a computation with the transformation law alone: the zero of $σ$ at the lattice point $1$ is inherited from the zero at $0$ through $σ (z + 1) = - \exp (η_{1} (z + 1 / 2)) σ (z)$ , and taking its difference quotient at $z = 0$ gives the derivative at $1$ , using continuity of the exponential from its defining series. The residue statement is the local form of $ζ = σ^{'} / σ$ at a simple zero of $σ$ , which is how the normalization $σ^{'} (0) = 1$ enters. This is the concrete display of the lattice-zero convention used in Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions.

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Sources

NIST Digital Library of Mathematical Functions, §23.2 and §23.5 (standard reference, not scraped)
J. S. Milne, Modular Functions and Modular Forms, Ch. 3, pp. 41-47 (standard reference, not scraped)
C. T. McMullen, Advanced Complex Analysis, Math 213a course notes, Ch. 5 §5.1, pp. 79-90 (standard reference, not scraped)