Jacobi sums and the Hilbert symbol for a power of two

Hiroo MIKI

doi:10.2969/jmsj/04820367

Hiroo MIKI

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JOURNAL FREE ACCESS

1996 Volume 48 Issue 2 Pages 367-408

DOI https://doi.org/10.2969/jmsj/04820367

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Published: 1996 Received: September 09, 1992 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) E. Artin and H. Hasse, Die beiden Ergänzungssätze zum Reziprozitätsgesetz der lⁿ-ten Potenzreste im Körper der lⁿ-ten Einheitswurzeln, Abh. Math. Sem. Univ. Hamburg, 6 (1928), 146-162.
2) R. Coleman and W. McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math., 385 (1988), 41-101.
3) R. Coleman, Anderson-Ihara theory: Gauss sums and circular units, Adv. Stud. Pure Math., 17 (1989), 55-72.
4) H. Hasse, Normenresttheorie galoisscher Zahlkörper mit Anwendungen auf Führer und Diskriminante abelscher Zahlkörper, J. Fac. Sci. Univ. Tokyo, 2 (1934), 477-498.
5) H. Hasse, Zetafunktionen und L-Funktionen zu einem arithmetischen Funktionen- körper vom Fermatschen Typus, Abh. Deut. Akad. Wiss. Berlin Kl. Math. Nat., 1954, n° 4 (1955), 70 pages.
6) Y. Ihara, Profinite braid groups, Galois representations and complex multiplications, Ann. of Math., 123 (1986), 43-106.
7) K. Iwasawa, On explicit formulas for the norm residue symbol, J. Math. Soc. Japan, 20 (1968), 151-165.
8) K. Iwasawa, Lectures on p-adic L-functions, Ann. of Math. Stud., 74, Princeton University Press, 1972.
9) A. Kudo, On Iwasawa's explicit formula for the norm residue symbol, Mem. Fac. Sci. Kyushu Univ., 26 (1971), 139-148.
10) E. Maus, Arithmetisch disjunkte Körper, J. Reine Angew. Math., 226 (1967), 184- 203.
11) H. Miki, On the l-adic expansion of certain Gauss sums and its applications, Adv. Stud. Pure Math., 12 (1987), 87-118.
12) H. Miki, On the conductor of the Jacobi sum Hecke character, Preprint, April 1988, 10 pages (unpublished): talk at the Number Theory Seminar (Goldfeld), Columbia Univ., Mar. 21, 1988.
13) H. Miki, On the conductor of the Jacobi sum Hecke character, Compositio Math., 92 (1994), 23-41.
14) H. Miki, On the calculation of certain Hilbert norm residue symbols and its application, J. Number Theory, 50 (1995), 87-105.
15) D. Rohrlich, Jacobi sums and explicit reciprocity laws, Compositio Math., 60 (1986), 97-114.
16) J.-P. Serre, Corps locaux, 2nd ed., Hermann, Paris, 1968, (English translation), Local Fields, GTM 67, Springer, New York-Berlin-Heidelberg, 1979.
17) L. Washington, Introduction to cyclotomic fields, GTM 83, Springer, Berlin-Heidelberg- New York, 1982.
18) A. Weil, Jacobi sums as “Grössencharaktere”. Trans. Amer. Math. Soc., 73 (1952), 487-495.

Right : [1] E. Artin and H. Hasse, Die beiden Ergänzungssätze zum Reziprozitätsgesetz der lⁿ-ten Potenzreste im Körper der lⁿ-ten Einheitswurzeln, Abh. Math. Sem. Univ. Hamburg, 6 (1928), 146-162.
[2] R. Coleman and W. McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math., 385 (1988), 41-101.
[3] R. Coleman, Anderson-Ihara theory: Gauss sums and circular units, Adv. Stud. Pure Math., 17 (1989), 55-72.
[4] H. Hasse, Normenresttheorie galoisscher Zahlkörper mit Anwendungen auf Führer und Diskriminante abelscher Zahlkörper, J. Fac. Sci. Univ. Tokyo, 2 (1934), 477-498.
[5] H. Hasse, Zetafunktionen und L-Funktionen zu einem arithmetischen Funktionen- körper vom Fermatschen Typus, Abh. Deut. Akad. Wiss. Berlin Kl. Math. Nat., 1954, n° 4 (1955), 70 pages.
[6] Y. Ihara, Profinite braid groups, Galois representations and complex multiplications, Ann. of Math., 123 (1986), 43-106.
[7] K. Iwasawa, On explicit formulas for the norm residue symbol, J. Math. Soc. Japan, 20 (1968), 151-165.
[8] K. Iwasawa, Lectures on p-adic L-functions, Ann. of Math. Stud., 74, Princeton University Press, 1972.
[9] A. Kudo, On Iwasawa's explicit formula for the norm residue symbol, Mem. Fac. Sci. Kyushu Univ., 26 (1971), 139-148.
[10] E. Maus, Arithmetisch disjunkte Körper, J. Reine Angew. Math., 226 (1967), 184-203.
[11] H. Miki, On the l-adic expansion of certain Gauss sums and its applications, Adv. Stud. Pure Math., 12 (1987), 87-118.
[12] H. Miki, On the conductor of the Jacobi sum Hecke character, Preprint, April 1988, 10 pages (unpublished): talk at the Number Theory Seminar (Goldfeld), Columbia Univ., Mar. 21, 1988.
[13] H. Miki, On the conductor of the Jacobi sum Hecke character, Compositio Math., 92 (1994), 23-41.
[14] H. Miki, On the calculation of certain Hilbert norm residue symbols and its application, J. Number Theory, 50 (1995), 87-105.
[15] D. Rohrlich, Jacobi sums and explicit reciprocity laws, Compositio Math., 60 (1986), 97-114.
[16] J. -P. Serre, Corps locaux, 2nd ed., Hermann, Paris, 1968, (English translation), Local Fields, GTM 67, Springer, New York-Berlin-Heidelberg, 1979.
[17] L. Washington, Introduction to cyclotomic fields, GTM 83, Springer, Berlin-Heidelberg- New York, 1982.
[18] A. Weil, Jacobi sums as “Grössencharaktere”. Trans. Amer. Math. Soc., 73 (1952), 487-495.

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