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For a given positive integer $d$ such that $-d$ is a fundamental discriminant, put $h(-d)$ for the cardinality of the ideal class group of the imaginary quadratic field $K_d = \mathbb{Q}(\sqrt{-d})$, and $C(-d)$ for the ideal class group.

For each prime $p \in \mathbb{Z}$ which splits in $K_d$, there is a pair $(c, c^{-1})$ of classes associated to $p$, namely the classes to the prime ideals $\mathfrak{p}, \overline{\mathfrak{p}}$ such that $(p) = \mathfrak{p} \overline{\mathfrak{p}}$. In general, a square-free positive integer $m$ with the property that each prime divisor of $m$ splits in $K_d$ can be attached to many possible classes, corresponding to how one factors the principal ideal $(m)$ in $\mathcal{O}_{K_d}$.

By Gauss composition laws, the ideal classes in $C(-d)$ canonically correspond to $\text{SL}_2(\mathbb{Z})$-equivalence classes of binary quadratic forms of discriminant $-d$.

My question is motivated by this question: Equidistribution of CM points in the principal genus

In particular, suppose that $G$ is a subgroup of $h(-d)$ of small index, say bounded by $O_\varepsilon(d^{\varepsilon})$ for any $\varepsilon > 0$. Let $\mathcal{G}$ be the corresponding subgroup of $\text{SL}_2(\mathbb{Z})$-equivalence classes of binary quadratic forms of discriminant $-d$. Is the set

$$\displaystyle \{m \leq x : m \text{ representable by a class in } \mathcal{G}\}$$

distributed as one should expect (that is, the proportion of representable integers should be $|\mathfrak{G}|/h(-d)$), even in small intervals?

Notice that the set of representable integers of a single binary quadratic form in a short interval can be very badly behaved; see for example: https://projecteuclid.org/euclid.dmj/1161093265. However, the set of numbers representable by any form of a fixed discriminant is much better behaved and one can expect the set of representable numbers in a short interval to be non-empty even when the interval has length $O_\varepsilon \left(d^{\varepsilon}\right)$.

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    $\begingroup$ By class field theory, isn't this really a question about the Chebotarev Density Theorem 'even in small intervals'? $\endgroup$
    – Stopple
    Aug 21, 2019 at 16:32
  • $\begingroup$ You might want to add the tags [class-field-theory] and, maybe, [complex-multiplication]. $\endgroup$
    – Wolfgang
    Aug 21, 2019 at 16:45
  • $\begingroup$ @Stopple I think what you said makes sense for primes... but does it still make sense for representable integers? $\endgroup$ Aug 21, 2019 at 17:55
  • $\begingroup$ I was implicitly thinking the question was about primes - I don't know the answer to your question. $\endgroup$
    – Stopple
    Aug 21, 2019 at 21:26

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