Math 213a: Complex analysis Problem Set #10 (8 December 2003): Doubly-periodic functions

1. Let L ⊂ C be a lattice, F its field of doubly-periodic functions, z_{j} (for j = 1, …, n) distinct points in C/L, and P_{j} ∈ P [X] polynomials with each P_{j}(0) = 0. We saw that if there exists f ∈ F with poles only at the z_{j} and principal part P_{j}(1/(z − z_{j})) at each z_{j} then ∑_{j=1}^{n} P′_{j}(0) = 0. Show that conversely if this one condition is satisfied then such f exists.

The next problem illustrates the “multiplicative” approach to elliptic functions. Fix q ∈ C^{∗} with |q| = 1. We saw that if q = exp(ω_{1}/ω_{2}) then the doubly-periodic functions for the lattice Zω_{1}⊕Zω_{2} amount to meromorphic functions on C^{∗}/q^{Z}, that is, meromorphic functions f on C^{∗} satisfying the identity f(qz) = f(z). Suppose more generally that f satisfies the identity f(qz) = cz^{n}f(z) for some c ∈ C^{∗} and n ∈ Z. While f no longer descends to a function on C^{∗}/q^{Z}, its zeros and poles are still invariant under multiplication by q, so we may speak of zeros and poles of f in C^{∗}/q^{Z}, each with a unique representative in a “period annulus” {z : |q| < |z| ≤ 1}.

2. i) Suppose these zeros and poles are z_{j} (j = 1, …, d) and z′_{j} (j = 1, …, d′), listed with multiplicity. Determine d − d′ and

∏_{j=1}^{d} z_{j} /∏_{j=1}^{d′} z′_{j} .

ii) Show that g(z) := ∏_{k=0}^{∞}(1 − q^{k}z) ∏_{k=1}^{∞}(1 − q^{k}z^{−1}) is an analytic function on C^{∗} that satisfies the identity g(z) = −z g(qz). Check that this is consistent with your answer to (i).

iii) Show that the function

∑_{m=−∞}^{∞} q^{(m^{2}−m)/2}(−z)^{m} = 1−z+q(z^{2}−z^{−1})+q^{3}(z^{−2}−z^{3})+q^{6}(z^{−4}−z^{3})+⋯

of z (an example of a theta function) satisfies the same identity as g, and deduce that this function equals cg for some c ∈ C^{∗} depending only on q.

iv) We shall see that c(q) = ∏_{k=1}^{∞}(1 − q^{k}), which is equivalent to the Jacobi triple product identity

∑_{m=∞}^{∞}(−1)^{m}q^{(m^{2}−m)/2}z^{m} = ∏_{k=0}^{∞}(1 − q^{k}) ∏_{k=0}^{∞}(1 − q^{k}z) ∏_{k=1}^{∞}(1 − q^{k}z^{−1}).

Assuming this, prove that

c(q) = 1 − q − q^{2} + q^{5} + q^{7} − q^{12} − + + − ⋯ = ∑_{m=−∞}^{∞}(−1)^{m}q^{m(3m+1)/2}

(Euler’s celebrated pentagonal-number identity), and that

c(q)^{3} = 1 − 3q + 5q^{3} − 7q^{6} + 9q^{10} − + ⋯ = ∑_{m=0}^{∞}(−1)^{m}(2m + 1)q^{(m^{2}+m)/2}.

More about the Weierstrass ℘-function and its differential equation:

3. Prove that

∑∑_{m,n∈Z}′ \frac{1}{(m + in)^{4}} = \frac{B(1/4, 1/2)^{4}}{240}

(where as usual the “′ ” in “∑ ∑′” indicates that the (m, n) = (0, 0) term is excluded from the sum). What is

∑∑_{m,n∈Z}′ \frac{1}{(m + ρn)^{6}}

where ρ = e^{2πi/3} is a cube root of unity? (Your answer should agree with the numerical value 5.8630 · · ·.)

4. i) Suppose L and M are lattices in C with L ⊆ M. Prove that for all z ∉ M we have ℘′_{M}(z) = ∑_{w} ℘′_{L}(z + w), where w ranges over (a set of representatives of) M/L.

ii) [2-isogeny] In the special case [M : L] = 2, express ℘_{M}(z) and ℘′_{M}(z) explicitly as rational functions of ℘_{L}(z), ℘′_{L}(z), and the root ℘_{L}(w) of P_{L}, where w ∈ M − L. Also, determine P_{M} in terms of ℘_{L}(w) and the coefficients of P_{L}. [The resulting identity connecting the elliptic integrals ∫ dx/\sqrt{P_{L}(x)} and ∫ du/\sqrt{P_{M}(u)} is known classically as “Landen’s transformation”.]

iii) For at least one of the lattices L = Z ⊕ Z\sqrt{−2}, L = Z ⊕ Z\sqrt{−3}, and L = Z ⊕ Z\sqrt{−4} = Z ⊕ 2iZ, use the formulas of (ii) to determine a^{3}/b^{2} where a, b are the coefficients of P_{L}(X) = 4X^{3} + aX + b. [These are rational numbers whose approximate numerical values are a^{3}/b^{2} = −34.44, −27.89, −27.16 respectively. It is known that in general a^{3}/b^{2} is an algebraic number for L = Z ⊕ Z\sqrt{−r} (r ∈ Q), or more generally L = Z ⊕ Zτ for τ ∈ H any quadratic irrational. These algebraic numbers a^{3}/b^{2} are in effect one of the topics of the theory of “complex multiplication”.]

This final problem set is due Monday, December 15, at the beginning of class.