## On the negative Pell equation.(English)Zbl 1230.11136

It is an interesting question for which fundamental discriminants $$D > 0$$ there is a solution to the negative Pell equation $$x^2 - dy^2 = -1$$ (here $$d = D/4$$ if $$d$$ is even and $$d=D$$ otherwise). Such a solution can only exist if $$D$$ has no prime divisor $$p \equiv 3$$ (mod 4), hence the number $$\mathcal{D}^-(X)$$ of such discriminants $$D \leq X$$ is $$O(X/\sqrt{\log X})$$. Considering discriminants with $$k$$ prime factors, Odoni showed the lower bound $$\mathcal{D}^{-}(X) \gg_k X (\log\log X)^k/\log X$$ which was improved by the reviwer to $$\gg X (\log X)^{-.62}$$. On the basis of a probablistic model, Stevenhagen conjectured $$\mathcal{D}^-(X) \sim \alpha X /\sqrt{\log X}$$ with $$\alpha = .58\ldots$$.
The authors make great progress on this conjecture by showing $.41 \frac{X}{\sqrt{\log X}} \leq \mathcal{D}^-(X) \leq .67\frac{X}{\sqrt{\log X}}$ for $$X$$ sufficiently large (ineffective!).
The starting point is the well-known criterion that the negative Pell equation is solvable if and only if for each $$k \geq 2$$ the $$2^k$$-rank of the class group equals the $$2^k$$-rank of the narrow class group of the underlying number field. The authors use this for $$k=2$$ (4-rank) to deduce their bounds. To this end, the authors study power moments of the type $\sum_{{D \leq X,\atop p \mid D \Rightarrow p \not\equiv 3\, (4)}} 2^{k \cdot \text{rk}_4(C_D)}$ and variants thereof. The asymptotic evaluation of these quantities requires very sophisticated algebraic and analytic tools that are too complicated to be described here.

### MSC:

 11R29 Class numbers, class groups, discriminants 11R11 Quadratic extensions 11E16 General binary quadratic forms

### Keywords:

Pell equation; class groups; Cohen-Lenstra heuristic
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### References:

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