Small discriminants of complex multiplication fields of elliptic curves over finite fields

Igor E. Shparlinski

doi:10.1007/s10587-015-0183-4

Small discriminants of complex multiplication fields of elliptic curves over finite fields

Igor E. Shparlinski^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves E over a prime finite field F_p of p elements, such that the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over F_p is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I.E. Shparlinski (2007).

Original language	English
Pages (from-to)	381-388
Number of pages	8
Journal	Czechoslovak Mathematical Journal
Volume	65
Issue number	2
DOIs	https://doi.org/10.1007/s10587-015-0183-4
Publication status	Published - 26 Jun 2015
Externally published	Yes

Keywords

complex multiplication field
elliptic curve
Frobenius discriminant

Access to Document

10.1007/s10587-015-0183-4

Cite this

@article{56f649ee7b5745799697f5f17d6634e3,

title = "Small discriminants of complex multiplication fields of elliptic curves over finite fields",

abstract = "We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves E over a prime finite field Fp of p elements, such that the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over Fp is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I.E. Shparlinski (2007).",

keywords = "complex multiplication field, elliptic curve, Frobenius discriminant",

author = "Shparlinski, {Igor E.}",

year = "2015",

month = jun,

day = "26",

doi = "10.1007/s10587-015-0183-4",

language = "English",

volume = "65",

pages = "381--388",

journal = "Czechoslovak Mathematical Journal",

issn = "0011-4642",

publisher = "Springer, Springer Nature",

number = "2",

}

TY - JOUR

T1 - Small discriminants of complex multiplication fields of elliptic curves over finite fields

AU - Shparlinski, Igor E.

PY - 2015/6/26

Y1 - 2015/6/26

N2 - We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves E over a prime finite field Fp of p elements, such that the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over Fp is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I.E. Shparlinski (2007).

AB - We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves E over a prime finite field Fp of p elements, such that the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over Fp is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I.E. Shparlinski (2007).

KW - complex multiplication field

KW - elliptic curve

KW - Frobenius discriminant

UR - http://www.scopus.com/inward/record.url?scp=84934947545&partnerID=8YFLogxK

U2 - 10.1007/s10587-015-0183-4

DO - 10.1007/s10587-015-0183-4

M3 - Article

AN - SCOPUS:84934947545

SN - 0011-4642

VL - 65

SP - 381

EP - 388

JO - Czechoslovak Mathematical Journal

JF - Czechoslovak Mathematical Journal

IS - 2

ER -

Small discriminants of complex multiplication fields of elliptic curves over finite fields

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this