Research PaperNo Access

Department of Mathematics and Informatics, Graduate School of Science, Chiba University, Yayoicho 1-33, Inage, Chiba 263-8522, Japan

https://doi.org/10.1142/S1793042125500393Cited by:0 (Source: Crossref)

View article

Abstract

In this paper, we express special values of the L-functions of certain CM elliptic curves which are related to Fermat curves in terms of the special values of generalized hypergeometric functions by comparing Bloch’s element with Ross’s element in the motivic cohomology group.

Keywords:

AMSC: 19F27, 33C20

References

[1] M. Asakura, M. Chida and F. Brunault , A numerical approach toward the p-adic Beilinson conjecture for elliptic curves over $ℚ$ , Res. Math. Sci. 10(1) (2023) 11. Crossref, Web of Science, Google Scholar
[2] A. Beilinson , Higher regulators and values of L-functions, J. Soviet Math. 30 (1985) 2036–2070. Crossref, Google Scholar
[3] S. Bloch , Lectures on Algebraic Cycles, Duke University Mathematics Series, Vol. IV (Duke University, Mathematics Department, Durham, NC, 1980). Google Scholar
[4] S. Bloch , Higher Regulators, Algebraic K-theory, and Zeta Functions of Elliptic Curves, CRM Monograph Series, Vol. 11 (American Mathematical Society, Providence, RI, 2000). Google Scholar
[5] C. Deninger and K. Wingberg , On the Beilinson conjectures for elliptic curves with complex multiplication, in Beilinson’s Conjectures on Special Values of L-functions, Perspectives in Mathematics, Vol. 4 (Academic Press, Boston, MA, 1988), pp. 249–272. Crossref, Google Scholar
[6] T. Dokchitser, R. de Jeu and D. Zagier , Numerical verification of Beilinson’s conjecture for $K_{2}$ of hyperelliptic curves, Compos. Math. 142(2) (2006) 339–373. Crossref, Web of Science, Google Scholar
[7] A. B. Goncharov and A. M. Levin , Zagier’s conjecture on $L (E, 2)$ , Invent. Math. 132(2) (1998) 393–432. Crossref, Web of Science, Google Scholar
[8] B. H. Gross and D. E. Rohrlich , Some results on the Mordell–Weil group of the Jacobian of the Fermat curve, Invent. Math. 44(3) (1978) 201–224. Crossref, Google Scholar
[9] R. Ito , The Beilinson conjectures for CM elliptic curves via hypergeometric functions, Ramanujan J. 45(2) (2018) 433–449. Crossref, Web of Science, Google Scholar
[10] LMFDB Collab., The L-functions and modular forms database (2022), http://www.lmfdb.org. Google Scholar
[11] A. Mellit , Elliptic dilogarithms and parallel lines, J. Number Theory 204 (2019) 1–24. Crossref, Web of Science, Google Scholar
[12] J. Nekovář , Beilinson’s conjectures, in Motives (Seattle, WA, 1991), Part 1, Proceedings of Symposia in Pure Mathematics, Vol. 55 (American Mathematical Society, Providence, RI, 1994), pp. 537–570. Crossref, Google Scholar
[13] N. Otsubo , On the regulator of Fermat motives and generalized hypergeometric functions, J. Reine Angew. Math. 660 (2011) 27–82. Web of Science, Google Scholar
[14] N. Otsubo , Certain values of Hecke L-functions and generalized hypergeometric functions, J. Number Theory 131(4) (2011) 648–660. Crossref, Web of Science, Google Scholar
[15] N. Otsubo , On the Abel–Jacobi maps of Fermat Jacobians, Math. Z. 270(1–2) (2012) 423–444. Crossref, Web of Science, Google Scholar
[16] N. Otsubo , On special values of Jacobi-sum Hecke L-functions, Exp. Math. 24(2) (2015) 247–259. Crossref, Web of Science, Google Scholar
[17] N. Otsubo , Homology of the Fermat tower and universal measures for Jacobi sums, Canad. Math. Bull. 59(3) (2016) 624–640. Crossref, Web of Science, Google Scholar
[18] M. Rogers and W. Zudilin , From L-series of elliptic curves to Mahler measures, Compos. Math. 148(2) (2012) 385–414. Crossref, Web of Science, Google Scholar
[19] D. E. Rohrlich , On the periods of abelian integrals and a formula of Chowla and Selberg, Invent. Math. 45(2) (1978) 193–211, Appendix to: B. H. Gross. Crossref, Google Scholar
[20] R. Ross , $K_{2}$ of Fermat curves with divisorial support at infinity, Compos. Math. 91(3) (1994) 223–240. Web of Science, Google Scholar
[21] S. Rosset and J. Tate , A reciprocity law for $K_{2}$ -traces, Comment. Math. Helv. 58(1) (1983) 38–47. Crossref, Google Scholar