Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {F}_d(X;G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">F</mi> <mi>d</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the set of number fields of degree d with absolute discriminant no larger than X and Galois group G. This set is known to be finite for any finite permutation group G and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$X \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we give a lower bound for the cases <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$G={\text {GL}}_2(\mathbb {F}_\ell )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mi mathvariant="normal">GL</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>ℓ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\text {PGL}}_2(\mathbb {F}_\ell )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">PGL</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>ℓ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for primes <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\ell \ge 13$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≥</mo> <mn>13</mn> </mrow> </math></EquationSource> </InlineEquation>.

Lower bounds for ${\text {GL}}_2(\mathbb {F}_\ell )$ number fields

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发表日期：

2025年10月18日

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Vittoria Cristante

摘要

Let $\mathcal {F}_d(X;G)$ $F_{d} (X; G)$ denote the set of number fields of degree d with absolute discriminant no larger than X and Galois group G. This set is known to be finite for any finite permutation group G and $X \ge 1$ $X \geq 1$ . In this paper, we give a lower bound for the cases $G={\text {GL}}_2(\mathbb {F}_\ell )$ $G = {GL}_{2} (F_{ℓ})$ and ${\text {PGL}}_2(\mathbb {F}_\ell )$ ${PGL}_{2} (F_{ℓ})$ for primes $\ell \ge 13$ $ℓ \geq 13$ .

Lower bounds for \({\text {GL}}_2(\mathbb {F}_\ell )\) number fields

摘要