<p>We study the Northcott and Bogomolov property for special values of Dedekind <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation>-functions at real values <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sigma \in \mathbbm {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. We prove, in particular, that the Bogomolov property is not satisfied when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma \ge \frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>≥</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. If <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sigma > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we produce certain families of number fields having arbitrarily large degrees, whose Dedekind <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation>-functions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\zeta _K(s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mi>K</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> attain arbitrarily small values at <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$s = \sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mi>σ</mi> </mrow> </math></EquationSource> </InlineEquation>. On the other hand, if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\frac{1}{2} \le \sigma \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>≤</mo> <mi>σ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we construct suitable families of quadratic number fields, employing either Soundararajan’s resonance method, which works when <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\frac{1}{2} \le \sigma < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>≤</mo> <mi>σ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, or results on random Euler products by Granville and Soundararajan, and by Lamzouri, which work when <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\frac{1}{2} < \sigma \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo><</mo> <mi>σ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We complete the study by proving that the Dedekind <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation> function together with the degree satisfies the Northcott property for every complex <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$s\in {\mathbbm {C}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\textrm{Re}(s) <0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Re</mtext> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, generalizing previous work of Généreux and Lalín.</p>

On Diophantine properties for values of Dedekind zeta functions

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发表日期：

2026年2月18日

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Jerson Caro,
Fabien Pazuki,
Riccardo Pengo

摘要

We study the Northcott and Bogomolov property for special values of Dedekind $\zeta $ $ζ$ -functions at real values $\sigma \in \mathbbm {R}$ $σ \in R$ . We prove, in particular, that the Bogomolov property is not satisfied when $\sigma \ge \frac{1}{2}$ $σ \geq \frac{1}{2}$ . If $\sigma > 1$ $σ > 1$ , we produce certain families of number fields having arbitrarily large degrees, whose Dedekind $\zeta $ $ζ$ -functions $\zeta _K(s)$ $ζ_{K} (s)$ attain arbitrarily small values at $s = \sigma $ $s = σ$ . On the other hand, if $\frac{1}{2} \le \sigma \le 1$ $\frac{1}{2} \leq σ \leq 1$ , we construct suitable families of quadratic number fields, employing either Soundararajan’s resonance method, which works when $\frac{1}{2} \le \sigma < 1$ $\frac{1}{2} \leq σ < 1$ , or results on random Euler products by Granville and Soundararajan, and by Lamzouri, which work when $\frac{1}{2} < \sigma \le 1$ $\frac{1}{2} < σ \leq 1$ . We complete the study by proving that the Dedekind $\zeta $ $ζ$ function together with the degree satisfies the Northcott property for every complex $s\in {\mathbbm {C}}$ $s \in C$ such that $\textrm{Re}(s) <0$ $Re (s) < 0$ , generalizing previous work of Généreux and Lalín.