# Goldfeld resolution of the quadratic class number problem

Goldfeld proved the following result. Let $$E$$ be an elliptic curve (with conductor $$N$$) over $$\mathbb{Q}$$ whose Hasse-Weil L-function has a zero at $$s = 1$$ with multiplicity $$g$$ then for sufficiently large $$D > 0$$ then the class number $$h(D)$$ of an imaginary quadratic field with discriminant $$-D$$ satisfies the following inequality if $$(N, D) = 1$$, $$h(D) > c(g, N) {\log(D)}^{g - \mu - 1} \exp(-21\sqrt{g\log(\log(D))})$$ where $$\mu = 1, 2$$ depending on $$\chi_{D}(-N) = {(-1)}^{g - \mu}$$, $$\chi_{D}$$ being the quadratic character with conductor $$-D$$.

Now, Goldfeld claims that if there is an elliptic curve with $$g = 3$$ (which later Gross - Zagier came up with), then we can effectively find bounds for $$D$$ that satisfy $$h(D) = n$$ for any fixed $$n$$. What I am unable to understand is the fact that suppose with $$g = 3$$ it happened to be that $$\mu = 2$$, the above stated lower bound on $$h(D)$$ would we decreasing with $$D$$ and we wouldn't be able to bound $$D$$ (from above). For a fixed elliptic curve $$\mu = \mu(D)$$ is a function of $$D$$. So my question is, in that case how is it possible to conclude anything useful when $$\mu = 2$$ i.e. when $$\chi_D(-N) = -1$$ with $$g = 3$$.

• Yeah so usually the auxiliary rank 3 curve that’s used has conductor equal to some small prime p times a square (see e.g. Prop. 7.4 of Gross-Zagier’s people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF01388809/… where p = 37). I see you assumed that $(D,N)= 1$ —- phew :p. (Otherwise see 4.3 of Oesterle’s numdam.org/article/SB_1983-1984__26__309_0.pdf .). Now $\chi_D(-1) = -1$ since we’re talking about im. q. fields, and if $\chi_D(N) = \chi_D(p) = 1$, then p would split, so p^h would be a nontrivial norm, so $h\gg \log{|D|}$. Otherwise $\mu = 1$ so use Goldfeld. Apr 25, 2021 at 0:26
• @alpoge: Can you turn your comment to a response? So that this question can be closed. Apr 25, 2021 at 6:50