| Title:
|
Small discriminants of complex multiplication fields of elliptic curves over finite fields (English) |
| Author:
|
Shparlinski, Igor E. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
2 |
| Year:
|
2015 |
| Pages:
|
381-388 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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). (English) |
| Keyword:
|
elliptic curve |
| Keyword:
|
complex multiplication field |
| Keyword:
|
Frobenius discriminant |
| MSC:
|
11G20 |
| MSC:
|
11N32 |
| MSC:
|
11R11 |
| idZBL:
|
Zbl 06486954 |
| idMR:
|
MR3360434 |
| DOI:
|
10.1007/s10587-015-0183-4 |
| . |
| Date available:
|
2015-06-16T17:46:13Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144277 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[5] Konyagin, S. V., Shparlinski, I. E.: Quadratic non-residues in short intervals.(to appear) in Proc. Amer. Math. Soc. |
| Reference:
|
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| Reference:
|
[7] Luca, F., Shparlinski, I. E.: Discriminants of complex multiplication fields of elliptic curves over finite fields.Can. Math. Bull. 50 (2007), 409-417. Zbl 1146.11034, MR 2344175, 10.4153/CMB-2007-039-2 |
| Reference:
|
[8] Montgomery, H. L.: Topics in Multiplicative Number Theory.Lecture Notes in Mathematics 227 Springer, Berlin (1971). Zbl 0216.03501, MR 0337847 |
| Reference:
|
[9] Shparlinski, I. E.: Tate-{S}hafarevich groups and Frobenius fields of reductions of elliptic curves.Q. J. Math. 61 (2010), 255-263. Zbl 1196.11080, MR 2646088, 10.1093/qmath/hap001 |
| Reference:
|
[10] Silverman, J. H.: The Arithmetic of Elliptic Curves.Graduate Texts in Mathematics 106 Springer, New York (2009). Zbl 1194.11005, MR 2514094 |
| . |
