Stark–Heegner theorem

Remarks

Equivalently, the fundamental discriminants $D < 0$ with class number $1$ are $-3, -4, -7, -8, -11, -19, -43, -67, -163$. Class number $1$ means that the ring of integers is a principal ideal domain, or, equivalently for rings of integers, a unique factorisation domain. Gauss's work on binary quadratic forms led him to expect this list; the theorem settles the case $h = 1$ of the class number problem. Heegner gave a proof using modular functions and complex multiplication, which was at first not accepted. Baker, with linear forms in logarithms, and Stark, with a method close to Heegner's, later gave complete proofs.

Example: $\Q(\sqrt{-163})$ has class number $1$. A related fact is that $n^2 + n + 41$ is prime for $n = 0, 1, \ldots, 39$; this works because $163 = 4 \cdot 41 - 1$. By contrast $\Q(\sqrt{-5})$ has class number $2$, as $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ shows.

Going deeper

For a student

The main difficulty is to rule out one possible extra field with a huge discriminant. Lower bounds for class numbers of the form $h(-d) \to \infty$ were known, but they were not effective, so they could not exclude a single exception. The theorem is closely related to the near-integer $e^{\pi \sqrt{163}}$, which comes from the $j$-invariant at a point of complex multiplication.