A NEW ASPECT OF CHEBYSHEV'S BIAS FOR ELLIPTIC CURVES OVER FUNCTION FIELDS

被引：1

作者：

Kaneko, Ikuya ^{[1
]}

Koyama, Shin-Ya ^{[2
]}

机构：

[1] CALTECH, Div Phys Math & Astron, 1200 E Calif Blvd, Pasadena, CA 91125 USA

[2] Toyo Univ, Dept Biomed Engn, 2100 Kujirai, Kawagoe, Saitama 3508585, Japan

来源：

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY | 2023年

关键词：

Deep Riemann Hypothesis; Grand Riemann Hypothesis; Birch- Swinnerton-Dyer conjecture; L-functions; elliptic curves; function fields; Chebyshev bias;

D O I：

10.1090/proc/16461

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

. This work addresses the prime number races for non-constant elliptic curves E over function fields. We prove that if rank(E) > 0, then there exist Chebyshev biases towards being negative, and otherwise there exist Chebyshev biases towards being positive. The key input is the convergence of the partial Euler product at the centre, which follows from the Deep Riemann Hypothesis over function fields.

引用

页码：5059 / 5068

页数：10

共 50 条

[41] Points on elliptic curves over finite fields
Skalba, M
ACTA ARITHMETICA, 2005, 117 (03) : 293 - 301
[42] EXCEPTIONAL ELLIPTIC CURVES OVER QUARTIC FIELDS
Najman, Filip
INTERNATIONAL JOURNAL OF NUMBER THEORY, 2012, 8 (05) : 1231 - 1246
[43] QUOTIENTS OF ELLIPTIC CURVES OVER FINITE FIELDS
Achter, Jeffrey D.
Wong, Siman
INTERNATIONAL JOURNAL OF NUMBER THEORY, 2013, 9 (06) : 1395 - 1412
[44] GENERATORS OF ELLIPTIC CURVES OVER FINITE FIELDS
Shparlinski, Igor E.
Voloch, Jose Felipe
BULLETIN OF THE INSTITUTE OF MATHEMATICS ACADEMIA SINICA NEW SERIES, 2014, 9 (04): : 657 - 670
[45] QUASIFUNCTIONS ON ELLIPTIC CURVES OVER LOCAL FIELDS
ZIMMER, HG
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 1979, 307 : 221 - 246
[46] Legendre elliptic curves over finite fields
Auer, R
Top, J
JOURNAL OF NUMBER THEORY, 2002, 95 (02) : 303 - 312
[47] ON ELLIPTIC-CURVES OVER PSEUDOLOCAL FIELDS
ANDRIICUK, VI
MATHEMATICS OF THE USSR-SBORNIK, 1981, 38 (01): : 83 - 94
[48] Elliptic and hyperelliptic curves over supersimple fields
Martin-Pizarro, A
Pillay, A
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 2004, 69 : 1 - 13
[49] PAIRINGS IN ELLIPTIC CURVES OVER GLOBAL FIELDS
VVEDENSKII, ON
MATHEMATICS OF THE USSR-IZVESTIYA, 1978, 12 (02): : 225 - 246
[50] Orchards in elliptic curves over finite fields
Padmanabhan, R.
Shukla, Alok
FINITE FIELDS AND THEIR APPLICATIONS, 2020, 68

← 1 2 3 4 5 →