[Paper Review] A new look at Hecke's indefinite theta series

Alexander Polishchuk|ArXiv.org|Dec 1, 2000

Advanced Mathematical Identities1 references3 citations

TL;DR

This paper establishes the modularity of a class of q-series defined by doubly periodic functions on Z² with vanishing line sums, generalizing Hecke's indefinite theta series. It proves these series transform as weight-1 modular forms and links their linear relations to dihedral group actions on (Z/NZ)², providing a complete characterization of such modular forms via group-theoretic orbit structures.

ABSTRACT

We describe a family of $q$-series generating the space of weight 1 modular forms coming from indefinite binary quadratic forms and study linear relations between these series.

Motivation & Objective

• To generalize Theorem 3 of [3] on univalued triple Massey products by relating q-series with vanishing line sums to Hecke's indefinite theta series.
• To establish the modularity of q-series of the form ∑_{m,n≥0} f(m,n)q^{Q(m,n)} − ∑_{m,n<0} f(m,n)q^{Q(m,n)} under vanishing horizontal and vertical sum conditions.
• To characterize the space of weight-1 modular forms generated by such q-series via the action of dihedral groups on (Z/NZ)².
• To show that linear relations among these q-series correspond precisely to orbits of the dihedral group action on the support of f modulo N.

Proposed method

• Define q-series Θ_{Q,f} as the difference of sums over the first and third quadrants, constrained by vanishing sums along all horizontal and vertical lines.
• Use the functional equations f(Ax) = f(Bx) = −f(x) with matrices A = [[−1,p],[0,1]], B = [[1,0],[r,−1]] where p = −2b/a, r = −2b/c.
• Show that the condition Tr(AB) = −2 + rp ∈ Z implies rp = 4b²/(ac) ∈ Z, enabling reduction to integer matrices modulo N.
• Study the group G_N ⊂ GL₂(Z/NZ) generated by A and B, showing its structure depends only on rp mod N.
• Classify admissible orbits of G_N on (Z/NZ)², distinguishing symmetric and asymmetric orbits, especially for N = 3, 5, 7.
• Relate each G_N-orbit to a modular form via the associated f, and compute initial terms of Θ_{Q,f} to establish linear independence.

Experimental results

Research questions

• RQ1 How do q-series with vanishing line sums relate to Hecke’s indefinite theta series?
• RQ2 What conditions on f and Q ensure that Θ_{Q,f} is a weight-1 modular form?
• RQ3 How are the linear relations among such q-series encoded in the group action on (Z/NZ)²?
• RQ4 What role does the value of rp mod N play in determining the structure of the group G_N and its orbits?
• RQ5 Can all weight-1 modular forms arising from this construction be realized as theta series from G_N-orbits?

Key findings

• The q-series Θ_{Q,f} is modular of weight 1 for a congruence subgroup of SL₂(Z), generalizing Hecke’s indefinite theta series.
• The space of such modular forms coincides exactly with the space generated by Hecke’s indefinite theta series.
• For N=3, the only admissible orbit is that of (1,0), and Θ_{Q,f} ≡ q^a + χ₃(r)q^c mod q^{a+1}, which vanishes only if r≡−1 mod 3 and a=c.
• For N=5, there are two linearly independent theta series when a≠c, with initial terms q^a±q^c or q^{4a}±q^{4c}, depending on p,r mod 5.
• For N=7, there are four linearly independent theta series: three from symmetric orbits and one from an asymmetric orbit, with distinct initial terms including q^{9a+c+6b}+q^{a+9c+6b}.
• The construction in [3] on elliptic curves corresponds to orbits of v_{s₁,s₂} = (b/a s₂−s₁, b/c s₁−s₂) mod N, with N=4D/(ac), and these are captured by the general framework.