Class numbers of binary quadratic polynomials
Zichen Yang
TL;DR
The paper addresses counting and bounding the proper class numbers $h^{+}(X)$ of shifted lattices associated to binary quadratic polynomials over number fields. It develops a mass-type class-number formula in dimension $2$ by computing local indices and correcting non-dyadic and dyadic $p$-adic cases, expressing $h^{+}(X)$ in terms of a base lattice, a conductor, and local factors. The main contributions include explicit formulas for the local densities $\beta^{+}_{\mathfrak{p}}(X;X)$, corrected computations for primes not dividing $2$, and a tractable dyadic case under mild divisibility assumptions, enabling a global expression $h^{+}(X)=\frac{h^{+}(L)}{[O^{+}(L):O^{+}(X)]}\cdot \frac{\beta_2^{+}(X)\,N_{K/\mathbb{Q}}(\mathfrak{M}_X)}{N_{K/\mathbb{Q}}(\gcd(\mathfrak{M}_X,\mathfrak{I}_X))}\prod_{\mathfrak{p}|\mathfrak{M}_X,\mathfrak{p}\nmid 2}\left(1-\frac{\eta(-\alpha)}{N(\mathfrak{p})}\right)$ (with appropriate local factors). Consequences include a lower bound $h^{+}(X)\gg N_{K/\mathbb{Q}}(\mathfrak{M}_X)^{1-\varepsilon}$ for totally real $K$, and finiteness results: for a fixed quadratic part and fixed $h^{+}$ there are only finitely many totally positive shifted lattices, enabling classification and counting of binary quadratic polynomials by their class numbers.
Abstract
In this paper, we give a formula for the proper class number of a binary quadratic polynomial assuming that the conductor ideal is sufficiently divisible at dyadic places. This allows us to study the growth of the proper class numbers of totally positive binary quadratic polynomials. As an application, we prove finiteness results on totally positive binary quadratic polynomials with a fixed quadratic part and a fixed proper class number.