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

Abstract

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 $\frac32$ 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 $\mathbb C F_0\otimesσ_0$ for one form $F_0$ and an irreducible $σ_0$, so the Petersson norm is $\|F\|^2=t_μ\|c_F(μ)\|^2$ for all cusp forms $F$, with one $t_μ>0$, $μ$ the least exponent of $F_0$ and $c_F(μ)$ the coefficient of $q^μ$ in $F$. We prove that the harmonic weak Maass forms of weight $\frac12$ in the $σ_0$-isotypic part with a pole of least possible order are exactly the forms $\mathcal H_F$ with principal part $q^{-μ}\overline{c_F(μ)}$, and $ξ_{1/2}\mathcal H_F=F/t_μ$. For the four theta lattices it is a theorem that $t_μ=\frac{s-1}{12}\sqrt s$, with $s$ the square-free part of $|\det L|/2$, and the $\mathcal H_F$ come from umbral moonshine. For the eight others we compute $F_0$, $t_μ$ and $\mathcal H_{F_0}$: the Hecke eigenvalues are integers and, at the primes computed, the $a_p$ of an elliptic curve, and for three of the eight $F_0$ 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 $g(X_0(4p)/w_p)-g(X_0(2p)/w_p)$ of the genera of two Atkin-Lehner quotients of modular curves. In a family $\mathcal D_p$ of discriminant forms, $p$ an odd prime, containing three of the eight, comparison with that classification shows that $\mathcal D_p$ is rigid exactly when $X_0(4p)/w_p$ has genus $1$ and $X_0(2p)/w_p$ has genus $0$.