Class Numbers of Imaginary Quadratic Orders in the Polynomial Hierarchy
Source: hexagonmath.org/abs/2611.00086v1
For $d \geq 3$ with $-d \equiv 0, 1 \pmod 4$, let $h(-d)$ be the number of classes of primitive positive definite binary quadratic forms of discriminant $-d$. We prove that $d \mapsto h(-d)$ is computable in deterministic polynomial time with an oracle for a language in the second level $\Sigma_2^{\text{P}}$ of the polynomial hierarchy. The proof uses only reduction theory of binary quadratic forms, Lagrange's theorem and Hermite normal forms; it does not assume the generalized Riemann hypothesis, does not take a generating set of the class group as input, and uses no randomness. A…
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