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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
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References:

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