Class numbers and integer points on some Pellian surfaces

Technical Summary: Class Numbers and Integer Points on Some Pellian Surfaces

Problem Statement
This paper investigates the distribution of nontrivial integer points on Pellian surfaces, specifically the surface defined by $t^2 - du^2 = 1$ within a bounded region. The central object of study is the counting function $N(B)$, which enumerates integer triples $(t, d, u)$ satisfying the equation with $\max(|t|, |d|, |u|) \le B$ and $u \neq 0$. The growth rate of $N(B)$ is intrinsically linked to the size of the fundamental solutions $\varepsilon_d$ of the Pell equation.

The paper addresses two primary questions:

1. What is the asymptotic behavior of $N(B)$?
2. What can be inferred about the typical size of fundamental solutions $\varepsilon_d$ and the average class numbers $h(d)$ for specific families of $d$, assuming conjectures regarding integer points on log K3 surfaces?

Methodology
The author employs a combination of analytic number theory, the geometry of numbers, and arithmetic geometry.

• Decomposition and Counting: For the unconditional upper bound of $N(B)$, the paper utilizes the "initial decomposition" of the Pell equation $(t+1)(t-1) = du^2$. This splits the problem into cases based on the parity of $t$ and divisibility of $d$. The author applies results from Fouvry and Jouve and Reuss, utilizing the approximate determinant method to bound the number of solutions to the resulting auxiliary equations $d_1 u_1^2 - d_2 u_2^2 = \theta$.
• Log K3 Surfaces and A1-curves: The paper shifts focus to the surface $U$ defined by $t^2 - (z^2+3)u^2 = 1$ (or equivalently $2uyz = y^2 - 3u^2 - 1$). The author identifies that integer points on the original Pellian surface often lie on "A1-curves" (rational curves defined over $\mathbb{Z}$). By removing these points, one studies the subset $U^\circ$.
• Conjectural Framework: The analysis relies heavily on Conjecture 1.2 by Browning and Wilsch, which predicts that for a smooth log K3 surface $U$, the number of integer points outside A1-curves grows logarithmically: $N^\circ_U(B) \ll (\log B)^{\rho_U + b}$, where $\rho_U$ is the Picard number and $b$ is the number of real components of the boundary divisor.
• Geometric Computation: To apply the Browning-Wilsch conjecture, the author explicitly computes the invariants of the surface $U$ defined by $k=3$. This involves resolving singularities (specifically an $A_2$ singularity) to determine the Picard number $\rho_U = 2$ and the boundary component count $b=2$, yielding an exponent of 4.
• Class Number Formulas: The paper connects the size of fundamental units to class numbers via the Dirichlet class number formula: $h(d) \log \varepsilon_d = \sqrt{d} L_d(1)$.

Key Contributions and Results

1. Bounds on Integer Points $N(B)$:

   • Theorem 1.1: The author establishes the lower bound $N(B) \gg B^{1/2}(\log B)^2$, derived from Hooley's theorem on the density of fundamental solutions.
   • Unconditional Upper Bound: Without assuming unproven conjectures, the paper proves $N(B) \ll_\epsilon B^{7/12 + \epsilon}$. This improves upon previous unconditional estimates by combining the initial decomposition with the approximate determinant method.
   • Conditional Upper Bound: Assuming Hooley's Conjecture (1.7) for $\alpha = 3/2$, the author shows $N(B) \ll B^{1/2}(\log B)^2$, matching the lower bound.

2. Lower Bounds for Fundamental Solutions:

   • Theorem 1.3: Assuming the Browning-Wilsch Conjecture for the specific log K3 surface $U$, the paper proves that for almost all $z$ (where $3 \nmid z$ and $z^2+3$ is square-free), the fundamental solution satisfies $\log \varepsilon_{d(z)} \gg d(z)^{1/8 - \epsilon}$.
   • This result is derived by showing that if the fundamental solutions were smaller, the number of integer points on the surface would exceed the logarithmic bound predicted by the conjecture.
   • Corollary 1.5: This implies the existence of infinitely many real quadratic fields $K$ with discriminant $\Delta_K$ such that $\log \varepsilon_K \gg \Delta_K^{1/8 - \epsilon}$.

3. Average of Class Numbers:

   • Theorem 1.6: Assuming the Browning-Wilsch Conjecture, the paper provides an improved upper bound for the average of class numbers $h(z^2+3)$ over square-free values where $3 \nmid z$. Specifically:
     $$\sum_{\substack{z \le Z \\ 3 \nmid z \\ \mu^2(z^2+3)=1}} h(z^2+3) \ll Z^{9/5} (\log Z)^{3/5}$$
   • This improves upon the trivial bound derived from Yamamoto's theorem, which yields a bound of order $Z^2 (\log Z)^{-2}$ for the same sum.

Significance and Claims
The paper claims to bridge the gap between the arithmetic of Pell equations and the geometry of log K3 surfaces.

• Reverse Engineering: A primary contribution is the "reverse process" of using conjectures about the scarcity of integer points on log K3 surfaces (specifically those not lying on A1-curves) to deduce lower bounds on the size of fundamental units.
• Refinement of Bounds: The work provides the first unconditional upper bound of $B^{7/12+\epsilon}$ for the counting function $N(B)$, refining the understanding of how integer points distribute on these surfaces.
• Conditional Improvements: By assuming the Browning-Wilsch conjecture, the paper demonstrates that fundamental solutions for the family $d=z^2+3$ are significantly larger than previously known for square-free $d$, and that the average class number for this family grows slower than the standard $Z^2$ rate.

The author notes that while the results for fundamental solutions are conditional on the Browning-Wilsch conjecture, the unconditional bounds on $N(B)$ are rigorous. The paper does not claim to prove the Browning-Wilsch conjecture itself but rather utilizes it as a tool to extract arithmetic information that is currently out of reach by standard analytic methods alone.