Group order formulas for reductions of CM elliptic curves

Silverberg, A.

doi:10.1090/conm/521/10277

Cited by 7 publications

(9 citation statements)

References 26 publications

(50 reference statements)

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“…The splitting of the Frobenius polynomial of E is well known in this case, with early results dating back to Gauss. The two statements below follow easily from [59,Th. 2.5,2.6].…”

Section: B1 the Ordinary Casementioning

confidence: 99%

Computing isomorphisms and embeddings of finite fields

Brieulle

Feo

Doliskani

et al. 2019

ACM Commun. Comput. Algebra

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Let F q be a finite field. Given two irreducible polynomials f, g over F q , with deg f dividing deg g, the finite field embedding problem asks to compute an explicit description of a field embedding of. When deg f = deg g, this is also known as the isomorphism problem.This problem, a special instance of polynomial factorization, plays a central role in computer algebra software. We review previous algorithms, due to Lenstra, Allombert, Rains, and Narayanan, and propose improvements and generalizations. Our detailed complexity analysis shows that our newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better.We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as open source. Our experiments show that our new variants consistently outperform available software.

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“…The splitting of the Frobenius polynomial of E is well known in this case, with early results dating back to Gauss. The two statements below follow easily from [59,Th. 2.5,2.6].…”

Section: B1 the Ordinary Casementioning

confidence: 99%

Computing isomorphisms and embeddings of finite fields

Brieulle

Feo

Doliskani

et al. 2019

ACM Commun. Comput. Algebra

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“…It is worth to mention that K. Rubin and A. Silverberg [RS09] generalized Stark's work to the case of all CM elliptic curves over arbitrary number fields. See [Si10] for a nice survey of their joint work with reasonable backgrounds. There were some partial generalizations before that, for instance W. Miller [Mi98] treated the case with CM by an imaginary quadratic field whose discriminant is even and not divisible by 3.…”

Section: Image Of Galois Representationsmentioning

confidence: 99%

Lang--Trotter Conjecture for CM Elliptic Curves

Wan

2021

Preprint

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Notation vii Chapter 1. Backgrounds: history, conjectures and theorems 1.1. Elliptic curves over rationals and finite fields 1.2. Sato-Tate and Lang-Trotter conjectures 1.3. Our main results Chapter 2. The Hardy-Littlewood conjecture and upper bound for π E,r (x) 2.1. The Hardy-Littlewood conjecture and sieve methods 2.2. Elliptic curves with complex multiplication 2.3. Proof of Theorem 1.3 2.4. Fixed trace in imaginary quadratic fields Chapter 3. Power residues and laws of reciprocity 3.1. Quadratic residues and reciprocity 3.2. Quartic residues and reciprocity 3.3. An incomplete character sum and quartic Gauß sums 3.4. Cubic residues and reciprocity 3.5. An incomplete character sum and cubic Gauß sums 3.6. Remarks Chapter 4. Complex multiplication by Q( √ −1) 4.1. Asymptotics for π E,r (x): conditional results 4.2. Initial transformations 4.3. Asymptotic evaluation of G d (x; r, α, β, γ) 4.4. From Conjecture 4.3 to Lang-Trotter Chapter 5. Complex multiplication by Q( √ −3) 5.1. Asymptotics for π E,r (x): conditional results 5.2. Initial transformations 5.3. Asymptotic evaluation of E d,k,ǫ (x; r, α, β, γ) 5.4. From Conjecture 5.5 to Lang-Trotter Chapter 6. Complex multiplication by remaining imaginary quadratic fields 6.1. Asymptotics for π E,r (x): conditional results 6.2. Initial transformations 6.3. Asymptotic evaluation of Π D (x; r, q, a) 6.4. From Conjecture 6.8 to Lang-Trotter: D = 2 6.5. From Conjecture 6.8 to Lang-Trotter: D 7 Chapter 7. Further interpretations on the Lang-Trotter constant iii iv CONTENTS 7.1. Positivity 7.2. Anomalous primes 7.3. Symmetry 7.4. Image of Galois representations Chapter 8. Historical notes and some perspectives 8.1. Gauß' last diary entry and Disquisitiones Arithmeticae 8.2. Variants and generalizations of Lang-Trotter conjecture Bibliography

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“…The theory of complex multiplication gives a = π u +π u for some π u ∈ O such that π uπu = q (see e.g. Theorem 14.16 in [3], [10, §II.10], or [8] for a thorough discussion). As K ⊂ k, we have K = kK, and we let ψ = ψ E/K be the Grössencharacter associated to E and K (see [10,§II.9] or [4]).…”

Section: Computing the Local Constantmentioning

confidence: 99%

“…If we choose f so that −D is not a square (mod f ), f is inert in K/Q, and so f O K 7 Determined up to the correspondence of class field theory. 8 See p.483 of [10], with f = 1 (in Silverman's notation), for a Weierstrauss equation. 9 Specifically with Sage's interface to John Cremona's 'mwrank' and Denis Simon's 'simon_two_descent.'…”

Section: Dihedral Extensions Of Qmentioning

confidence: 99%