# The Class Number One Problem for Real Quadratic Fields

An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick summary, but see Appendix A in Serre's book "Lectures on the Mordell-Weil Theorem" for more details:

Given an imaginary quadratic $K$ of class number 1, one considers an elliptic curve $E$ with CM by $O_K$, which is unique up to $\mathbb{C}$-isomorphism. Its $j$-invariant lies in $\mathbb{Z}$. Given any integer $n$, all of whose prime divisors are inert in $K$, our $E$ yields a unique integral point on $Y_{nonsplit}(n)$. That means, fixing an $n$ and letting $p$ be its largest prime divisor, any imaginary quadratic $K$ of class number 1 whose discriminant is larger than $4p$ in absolute value will furnish a unique integral point on $Y_{nonsplit}(n)$. Hence, determining the integral points on $Y_{nonsplit}(n)$ for one $n$ for which there are finitely many integral points solves the problem. Heegner used $n=24$, as did Stark.

My question is whether a similar approach has been, or can be, used in the real quadratic case;

the goal being to prove that there are infinitely many such fields of class number one.

For what it's worth, here's what would happen in my pipe dream:

1. To a real quadratic $K$ of class number 1 one attaches uniquely an abelian surface (or some other kind of object) with real multiplication by $O_K$;

2. One shows that this object gives rise to a unique rational point on some moduli space, and conversely, all such rational points would come from a $K$; ($Y_{nonsplit}$ plays this role in the imaginary case...)

3. One shows that the moduli space has infinitely many rational points.

(Since this is a well-known open problem, I'll follow the advice of the FAQ and make it community-wiki. I am a little embarrassed asking such a speculative question, but I also feel that speculation drives a lot of mathematical research.)