Schertz style class invariants for quartic CM fields

Enge, Andreas; Streng, Marco

doi:10.48550/arxiv.1610.04505

Cited by 2 publications

(2 citation statements)

References 13 publications

(24 reference statements)

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“…For any f ∈ Q(C), Theorem 4 of Schertz [30] states f (τ ) ∈ K O ∪ {∞} and gives the Gal(K O /K )-orbit as {g(N τ i ) : i}, under an additional condition on the function f (1/z). However, the condition on f (1/z) is not needed, as stated in Theorems 3.9 and 4.4 of [13]. This proves the result.…”

Section: Moreover For Any Such Choice We Havesupporting

confidence: 72%

Generalized class polynomials

Houben

Streng

2022

Res. number theory

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The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber’s functions, which reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Bröker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Bröker–Stevenhagen bound. We provide examples matching Weber’s reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2.

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Section: Moreover For Any Such Choice We Havesupporting

confidence: 72%

Generalized class polynomials

Houben

Streng

2022

Res. number theory

View full text Add to dashboard Cite

show abstract

“…Moreover, for any such choice, we have For any f ∈ Q(C), Theorem 4 of Schertz [31] states f (τ ) ∈ K O ∪ {∞} and gives the Gal(K O /K)-orbit as {g(N τ i ) : i}, under an additional condition on the function f (1/z). However, the condition on f (1/z) is not needed, as stated in Theorems 3.9 and 4.4 of [13]. This proves the result.…”

supporting

confidence: 72%

Generalized class polynomials

Houben¹,

Streng²

2022

Preprint

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The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber's functions, that reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Bröker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Bröker-Stevenhagen bound. We provide examples matching Weber's reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2.

show abstract

Schertz style class invariants for quartic CM fields

Cited by 2 publications

References 13 publications

Generalized class polynomials

Generalized class polynomials

Generalized class polynomials

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