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

Igor E. Shparlinski

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 2, page 381-388
  • ISSN: 0011-4642

Abstract

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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).

How to cite

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Shparlinski, Igor E.. "Small discriminants of complex multiplication fields of elliptic curves over finite fields." Czechoslovak Mathematical Journal 65.2 (2015): 381-388. <http://eudml.org/doc/270129>.

@article{Shparlinski2015,
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 $\mathbb \{F\}_p$ of $p$ elements, such that the discriminant $D(E)$ of the quadratic number field containing the endomorphism ring of $E$ over $\mathbb \{F\}_p$ 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).},
author = {Shparlinski, Igor E.},
journal = {Czechoslovak Mathematical Journal},
keywords = {elliptic curve; complex multiplication field; Frobenius discriminant},
language = {eng},
number = {2},
pages = {381-388},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Small discriminants of complex multiplication fields of elliptic curves over finite fields},
url = {http://eudml.org/doc/270129},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Shparlinski, Igor E.
TI - Small discriminants of complex multiplication fields of elliptic curves over finite fields
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 2
SP - 381
EP - 388
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 $\mathbb {F}_p$ of $p$ elements, such that the discriminant $D(E)$ of the quadratic number field containing the endomorphism ring of $E$ over $\mathbb {F}_p$ 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).
LA - eng
KW - elliptic curve; complex multiplication field; Frobenius discriminant
UR - http://eudml.org/doc/270129
ER -

References

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  1. Cojocaru, A. C., Questions about the reductions modulo primes of an elliptic curve, Number Theory; Papers from the 7th Conference of the Canadian Number Theory Association, University of Montreal, Canada, 2002. CRM Proc. Lecture Notes 36 American Mathematical Society, Providence (2004), 61-79 H. Kisilevsky et al. (2004) Zbl1085.11030MR2076566
  2. Cojocaru, A. C., Duke, W., 10.1007/s00208-004-0517-2, Math. Ann. 329 (2004), 513-534. (2004) Zbl1062.11039MR2127988DOI10.1007/s00208-004-0517-2
  3. Cojocaru, A. C., Fouvry, E., Murty, M. R., 10.4153/CJM-2005-045-7, Can. J. Math. 57 (2005), 1155-1177. (2005) Zbl1094.11021MR2178556DOI10.4153/CJM-2005-045-7
  4. Iwaniec, H., Kowalski, E., Analytic Number Theory, Amer. Math. Soc. Colloquium Publications 53 American Mathematical Society, Providence (2004). (2004) Zbl1059.11001MR2061214
  5. Konyagin, S. V., Shparlinski, I. E., Quadratic non-residues in short intervals, (to appear) in Proc. Amer. Math. Soc. 
  6. H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. Math. (2) 126 (1987), 649-673. (1987) Zbl0629.10006MR0916721
  7. Luca, F., Shparlinski, I. E., 10.4153/CMB-2007-039-2, Can. Math. Bull. 50 (2007), 409-417. (2007) Zbl1146.11034MR2344175DOI10.4153/CMB-2007-039-2
  8. Montgomery, H. L., Topics in Multiplicative Number Theory, Lecture Notes in Mathematics 227 Springer, Berlin (1971). (1971) Zbl0216.03501MR0337847
  9. Shparlinski, I. E., 10.1093/qmath/hap001, Q. J. Math. 61 (2010), 255-263. (2010) Zbl1196.11080MR2646088DOI10.1093/qmath/hap001
  10. Silverman, J. H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106 Springer, New York (2009). (2009) Zbl1194.11005MR2514094

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