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Let $D$ be a square-free positive integer which is the fundamental discriminant of a real quadratic field. Consider the following quadratic form $$Q_{D}(x,y)=x^2+Dy^2.$$ My questions are :

  • What is the density of the set $\{\ell \hbox{ prime}| Q_{D}(x,y)=\ell\quad\hbox{has a solution}\}$ if it is knowen ?
  • How can we use modular forms to answer such a question ? ( how to use theta series for example and distribution of eigenvlues of newforms ? )
  • Thanks for any comments !
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In general, modular forms are very useful for understanding the number and distribution of representations by an integral quadratic form in three or more variables, but for binary quadratic forms these kinds of problems are usually treated by more classical algebraic number theory.

I recommend you to study Cox's book "Primes of the form $x^2+ny^2$" which addresses your question. In particular, Theorem 9.2 tells us that a prime $\ell\nmid 2D$ is represented by $Q_D(x,y)$ if and only if $\ell$ splits completely in the ring class field $L$ of the order $\mathbb{Z}[\sqrt{-D}]$. This implies, e.g. by the Chebotarev density theorem, that the relative density of these primes is the reciprocal of $[L:\mathbb{Q}]=2h(-4D)$.

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    $\begingroup$ One might add that (among others) a reason that typical devices from modular forms are not so effective with binary theta series is that the weight is $1$, so the Fourier coefficients of cuspforms are not much smaller than those of Eisenstein series (nevermind the issue of holomorphy of the latter), so that Eisenstein series cannot reliably give a leading term in asymptotics, unlike quadratic forms in more variables, which lead to higher-weight modular forms (whether half-integral weight or not). $\endgroup$ – paul garrett Feb 19 '17 at 19:37

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