详细信息    查看全文

• 作者：Rupam Barman (1)
  Neelam Saikia (2)
  
• 关键词：Polynomial equation ; Characters of finite fields ; Gaussian hypergeometric series ; 11T21 ; 33C20 ; 11G20
• 刊名：The Ramanujan Journal
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：35
• 期：3
• 页码：427-441
• 全文大小：549 KB
• 参考文献：1. Ahlgren, S., Ono, K.: A Gaussian hypergeometric series and Apery number congruences. J. Reine Angew. Math. 518, 187-12 (2000)
  2. Barman, R., Kalita, G.: Hypergeometric functions and a family of algebraic curves. Ramanujan J. 28(2), 175-85 (2012) CrossRef
  3. Barman, R., Kalita, G.: Certain values of Gaussian hypergeometric series and a family of algebraic curves. Int. J. Number Theory 8(4), 945-61 (2012) CrossRef
  4. Barman, R., Kalita, G.: Hypergeometric functions over \(\mathbb{F}_{q}\) and traces of Frobenius for elliptic curves. Proc. Am. Math. Soc. 141, 3403-410 (2013) CrossRef
  5. Barman, R., Kalita, G.: Elliptic curves and special values of Gaussian hypergeometric series. J. Number Theory 133(9), 3099-111 (2013) CrossRef
  6. Barman, R., Kalita, G.: On the polynomial / x ^{/ d} + / ax+ / b over \(\mathbb{F}_{q}\) and Gaussian hypergeometric series. Int. J. Number Theory 9(7), 1753-763 (2013) CrossRef
  7. Frechette, S., Ono, K., Papanikolas, M.: Gaussian hypergeometric functions and traces of Hecke operators. Int. Math. Res. Not. 2004(60), 3233-262 (2004) CrossRef
  8. Fuselier, J.: Hypergeometric functions over \(\mathbb{F}_{p}\) and relations to elliptic curves and modular forms. Proc. Am. Math. Soc. 138, 109-23 (2010) CrossRef
  9. Greene, J.: Hypergeometric functions over finite fields. Trans. Am. Math. Soc. 301(1), 77-01 (1987) CrossRef
  10. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, New York (1990) CrossRef
  11. Koike, M.: Hypergeometric series over finite fields and Apéry numbers. Hiroshima Math. J. 22, 461-67 (1992)
  12. Lang, S.: Cyclotomic Fields I and II. Graduate Texts in Mathematics, vol. 121. Springer, New York (1990)
  13. Lennon, C.: Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves. Proc. Am. Math. Soc. 139, 1931-938 (2011) CrossRef
  14. Lennon, C.: Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold. J. Number Theory 131(12), 2320-351 (2011) CrossRef
  15. McCarthy, D.: ₃ / F ₂ Hypergeometric series and periods of elliptic curves. Int. J. Number Theory 6(3), 461-70 (2010) CrossRef
  16. Ono, K.: Values of Gaussian hypergeometric series. Trans. Am. Math. Soc. 350(3), 1205-223 (1998) CrossRef
  17. Ono, K.: Web of Modularity:
• 作者单位：Rupam Barman (1)
  Neelam Saikia (2)
  
  1. Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi, 110016, India
  2. Department of Mathematical Sciences, Tezpur University, Napaam, Sonitpur, Assam, 784028, India
  
• ISSN：1572-9303

文摘

Let i and d be integers such that d?, 0id, and i|d. We explicitly find the number of solutions of the polynomial equations x d +ax i +b=0 and x d +ax d?em class="a-plus-plus">i +b=0 over \(\mathbb{F}_{q}\) in terms of special values of \(_{\frac{d-i}{i}}F_{\frac{d-2i}{i}}\) and \(_{\frac{d}{i}}F_{\frac{d-i}{i}}\) , Gaussian hypergeometric series with characters of orders \(\frac{d}{i}-1\) and \(\frac{d}{i}\) as parameters. This solves the problem posed by Ken Ono (Web of modularity: arithmetic of the coefficients of modular forms and q-series, vol.?102, p.?204, 2004) on special values of Gaussian hypergeometric series n+1 F n for n>2.