Petersson-rigid lattices of signature (2,1) and their harmonic weak Maass forms

An even lattice L of signature $(2,1)$ is Petersson-rigid if the orthogonal group of its discriminant form acts absolutely irreducibly on the space of cusp forms of weight 32 for its Weil representation, a space of dimension at least two. We prove, with computer assistance, that there are exactly twelve such lattices up to isometry, and we determine their root bases. Four of them are theta lattices, whose cusp forms are lifts of unary theta series. On each, the space is C F₀⊗σ₀ for one form F₀ and an irreducible σ₀, so the Petersson norm is \|F\|²=t_μ\|cF(μ)\|² for all cusp forms F, with one t_μ>0, $μ$ the least exponent of F₀ and cF(μ) the coefficient of q^μ in F. We prove that the harmonic weak Maass forms of weight 12 in the σ₀-isotypic part with a pole of least possible order are exactly the forms HF with principal part q-μcF(μ)̄, and ξ1/2 HF=F/t_μ. For the four theta lattices it is a theorem that t_μ=s-1/12√ s, with s the square-free part of | L|/2, and the HF come from umbral moonshine. For the eight others we compute F₀, t_μ and HF₀: the Hecke eigenvalues are integers and, at the primes computed, the aₚ of an elliptic curve, and for three of the eight F₀ is explicit as an eta-theta product. We prove, for every prime $p>3$, a closed formula in p and one class number for the difference g(X₀(4p)/wₚ)-g(X₀(2p)/wₚ) of the genera of two Atkin-Lehner quotients of modular curves. In a family Dₚ of discriminant forms, p an odd prime, containing three of the eight, comparison with that classification shows that Dₚ is rigid exactly when X₀(4p)/wₚ has genus $1$ and X₀(2p)/wₚ has genus $0$.