For an elliptic curve E over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> without complex multiplication, the Lang–Trotter conjecture is that <Equation ID="Equ12"> <EquationSource Format="TEX">$ \# \{ p<X \mid E \text { has a supersingular reduction at } p \} \sim \frac{c \sqrt{X}}{\log X} $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>#</mo> <mrow> <mo stretchy="false">{</mo> <mi>p</mi> <mo><</mo> <mi>X</mi> <mo>∣</mo> <mi>E</mi> <mrow> <mi mathvariant="normal">has</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">supersingular</mi> <mi mathvariant="normal">reduction</mi> <mi mathvariant="normal">at</mi> </mrow> <mi mathvariant="normal">p</mi> <mo stretchy="false">}</mo> </mrow> <mo>∼</mo> <mfrac> <mrow> <mi mathvariant="normal">c</mi> <msqrt> <mi mathvariant="normal">X</mi> </msqrt> </mrow> <mrow> <mo>log</mo> <mi mathvariant="normal">X</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </Equation>as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$X \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$c>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a constant depending only on E. Fouvry and Murty obtained an average estimation related to the Lang–Trotter conjecture, called the Lang–Trotter conjecture on average. We consider the Lang–Trotter conjecture for curves of genus 2 and obtain a similar result to the Lang–Trotter conjecture on average for the family of curves <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C_{\lambda }:y^2=x(x-1)(x+1)(x-{\lambda })(x-1/ \lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>λ</mi> </msub> <mo>:</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>=</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. These curves are characterized as curves of genus two with a reduced automorphism group containing the Klein 4-group.

The Lang-Trotter conjecture on average for genus-2 curves with Klein-4 reduced automorphism group

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2025年9月4日

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Chihiro Ando

摘要

For an elliptic curve E over $\mathbb {Q}$ $Q$ without complex multiplication, the Lang–Trotter conjecture is that $ \# \{ p<X \mid E \text { has a supersingular reduction at } p \} \sim \frac{c \sqrt{X}}{\log X} $ $# {p < X ∣ E has a supersingular reduction at p} \sim \frac{c \sqrt{X}}{log X}$ as $X \rightarrow \infty $ $X \to \infty$ , where $c>0$ $c > 0$ is a constant depending only on E. Fouvry and Murty obtained an average estimation related to the Lang–Trotter conjecture, called the Lang–Trotter conjecture on average. We consider the Lang–Trotter conjecture for curves of genus 2 and obtain a similar result to the Lang–Trotter conjecture on average for the family of curves $C_{\lambda }:y^2=x(x-1)(x+1)(x-{\lambda })(x-1/ \lambda )$ $C_{λ} : y^{2} = x (x - 1) (x + 1) (x - λ) (x - 1 / λ)$ . These curves are characterized as curves of genus two with a reduced automorphism group containing the Klein 4-group.