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On a unit group generated by special values
of Siegel modular functions


Authors: T. Fukuda and K. Komatsu
Journal: Math. Comp. 69 (2000), 1207-1212
MSC (1991): Primary 11G15, 11R27, 11Y40
Published electronically: February 19, 1999
MathSciNet review: 1651753
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Abstract | References | Similar Articles | Additional Information

Abstract: There has been important progress in constructing units and $S$-units associated to curves of genus 2 or 3. These approaches are based mainly on the consideration of properties of Jacobian varieties associated to hyperelliptic curves of genus 2 or 3. In this paper, we construct a unit group of the ray class field $k_6$ of $\mathbb{Q}(\exp(2\pi i/5))$ modulo 6 with full rank by special values of Siegel modular functions and circular units. We note that $k_6=\mathbb{Q}(\exp(2\pi i/15),\,\sqrt[5]{-24}\,)$. Our construction of units is number theoretic, and closely based on Shimura's work describing explicitly the Galois actions on the special values of theta functions.


References [Enhancements On Off] (What's this?)

  • 1. Jane Arledge, 𝑆-units attached to genus 3 hyperelliptic curves, J. Number Theory 63 (1997), no. 1, 12–29. MR 1438645, 10.1006/jnth.1997.2073
  • 2. J. Boxall and E. Bavencoffe, Quelques propriétés arithmétiques des points de 3-division de la jacobienne de 𝑦²=𝑥⁵-1, Sém. Théor. Nombres Bordeaux (2) 4 (1992), no. 1, 113–128 (French, with French summary). MR 1183921
  • 3. David Grant, Units from 3- and 4-torsion on Jacobians of curves of genus 2, Compositio Math. 94 (1994), no. 3, 311–320. MR 1310862
  • 4. K. Komatsu, Construction of normal basis by special values of Siegel modular functions, submitted to Proc. Amer. Math. Soc.
  • 5. Goro Shimura, Theta functions with complex multiplication, Duke Math. J. 43 (1976), no. 4, 673–696. MR 0424705

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Additional Information

T. Fukuda
Affiliation: Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan
Email: fukuda@math.cit.nihon-u.ac.jp

K. Komatsu
Affiliation: Department of Information and Computer Science, School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169, Japan
Email: kkomatsu@mn.waseda.mse.jp

DOI: https://doi.org/10.1090/S0025-5718-99-01118-7
Keywords: Siegel modular functions, unit groups, computation
Received by editor(s): October 16, 1997
Received by editor(s) in revised form: August 14, 1998
Published electronically: February 19, 1999
Article copyright: © Copyright 2000 American Mathematical Society