We classify all instances of the condition <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a_{p}(f) \equiv x \bmod \lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mi>x</mi> <mspace width="0.277778em" /> <mo>mod</mo> <mspace width="0.277778em" /> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation> being related to a congruence on the prime p, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$a_{p}(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denotes the pth Fourier coefficient of a classical normalised cuspidal eigenform f and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> is a prime in the number field generated by the Fourier coefficients of f. This classification is done in terms of the (projective) image of the mod <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> Galois representation associated with f and extends work by Swinnerton-Dyer. We highlight that for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$x = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, this condition is more often implied by a congruence on the prime p than the general value of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$a_{p}(f) \bmod \lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.277778em" /> <mo>mod</mo> <mspace width="0.277778em" /> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation>. Finally, we illustrate various instances of these congruences through examples from the setting of weight 2 newforms attached to rational elliptic curves.

Congruence conditions for the mod $\lambda $ values of the Fourier coefficients of classical eigenforms

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2026年1月31日

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Michael A. Daas

摘要

We classify all instances of the condition $a_{p}(f) \equiv x \bmod \lambda $ $a_{p} (f) \equiv x mod λ$ being related to a congruence on the prime p, where $a_{p}(f)$ $a_{p} (f)$ denotes the pth Fourier coefficient of a classical normalised cuspidal eigenform f and $\lambda $ $λ$ is a prime in the number field generated by the Fourier coefficients of f. This classification is done in terms of the (projective) image of the mod $\lambda $ $λ$ Galois representation associated with f and extends work by Swinnerton-Dyer. We highlight that for $x = 0$ $x = 0$ , this condition is more often implied by a congruence on the prime p than the general value of $a_{p}(f) \bmod \lambda $ $a_{p} (f) mod λ$ . Finally, we illustrate various instances of these congruences through examples from the setting of weight 2 newforms attached to rational elliptic curves.

Congruence conditions for the mod \(\lambda \) values of the Fourier coefficients of classical eigenforms

摘要