The \(p\)-adic sigma function. (English) Zbl 0735.14020

The authors construct a \(p\)-adic analog of the Weierstrass sigma function for complex elliptic curves. More precisely, they construct a single- valued function on the formal group \(E^ f\) of an elliptic curve \(E\) defined over the field of fractions \(K\) of a complete discrete valuation ring \(R\), whose residue field is of characteristic \(p>0\). As in the classical case, this sigma function is — up to a constant — uniquely defined by the elliptic curve \(E\). One of the main results of the paper (theorem 3.1) says that the \(p\)-adic sigma function enjoys, and is in fact characterized by, any one of several properties which are analogs of those of the classical Weierstrass sigma function.
The whole construction is based on the assumption that the elliptic curve \(E\) is of ordinary reduction, i.e., over the algebraic closure of the residue field of \(R\), the formal group \(E^ f\) is isomorphic to the formal multiplicative group \(\mathbb{G}^ f_ m\). As for applications of the \(p\)-adic sigma function, the authors have already given some of them in two earlier papers, mainly with a view towards the theory of canonical heights for points on elliptic curves in characteristic \(p>0\) and \(p\)- adic analogs of the conjectures of Birch and Swinnerton-Dyer [cf. B. Mazur and J. Tate in Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. I: Arithmetic, Progr. Math. 35, 195-237 (1983; Zbl 0574.14036); B. Mazur, J. Tate and J. Teitelbaum, Invent. Math. 84, 1-48 (1986; Zbl 0699.14028)].
In this regard, the present paper provides the detailed construction of the \(p\)-adic sigma function, whose existence was already assumed (and, in fact, used) in a wider context. As the authors point out, there are many other approaches to construct \(p\)-adic sigma functions or, more generally, \(p\)-adic theta functions in the literature, according to the many different contexts and various purposes. The one given here is especially motivated by the connection with the theory of canonical heights.


14G20 Local ground fields in algebraic geometry
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14H52 Elliptic curves
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14K25 Theta functions and abelian varieties
Full Text: DOI


[1] I. Barsotti, Considerazioni sulle funzioni theta , Symposia Mathematica, Vol. III (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 247-277. · Zbl 0194.52201
[2] D. Bernardi, Hauteur \(p\)-adique sur les courbes elliptiques , Seminar on Number Theory, Paris 1979-80, Progr. Math., vol. 12, Birkhäuser Boston, Mass., 1981, pp. 1-14. · Zbl 0475.14034
[3] D. Bernardi, C. Goldstein, and N. Stephens, Notes \(p\)-adiques sur les courbes elliptiques , J. Reine Angew. Math. 351 (1984), 129-170. · Zbl 0529.14018
[4] N. Bourbaki, Éléments de mathématique. Fascicule XXVII. Algèbre commutative. Chapitre 1: Modules plats. Chapitre 2: Localisation , Actualités Scientifiques et Industrielles, No. 1290, Herman, Paris, 1961. · Zbl 0108.04002
[5] L. Breen, Fonctions thêta et théorème du cube , Lecture Notes in Mathematics, vol. 980, Springer-Verlag, Berlin, 1983. · Zbl 0558.14029
[6] V. Cristante, Theta functions and Barsotti-Tate groups , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 2, 181-215. · Zbl 0438.14027
[7] N. M. Katz, \(p\)-adic properties of modular schemes and modular forms , Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 69-190. Lecture Notes in Mathematics, Vol. 350. · Zbl 0271.10033
[8] S. Lang, Elliptic functions , 2nd ed. ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. · Zbl 0615.14018
[9] B. Mazur and J. Tate, Canonical height pairings via biextensions , Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 195-237. · Zbl 0574.14036
[10] B. Mazur, J. Tate, and J. Teitelbaum, On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer , Invent. Math. 84 (1986), no. 1, 1-48. · Zbl 0699.14028
[11] J. McCabe, \(P\)-adic theta functions , Ph.D. thesis, Harvard, 1968.
[12] W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes , Lecture Notes in Mathematics, vol. 264, Springer-Verlag, Berlin, 1972. · Zbl 0243.14013
[13] 1 H. Morikawa, Theta functions and abelian varieties over valuation fields of rank one. I , Nagoya Math. J. 20 (1962), 1-27. · Zbl 0115.39001
[14] 2 H. Morikawa, On theta functions and abelian varieties over valuation fields of rank one. II. Theta functions and abelian functions of characteristic \(p(>0)\). , Nagoya Math. J. 21 (1962), 231-250. · Zbl 0115.39002
[15] A. Néron, Hauteurs et fonctions thêta , Rend. Sem. Mat. Fis. Milano 46 (1976), 111-135 (1978). · Zbl 0471.14024
[16] A. Néron, Fonctions thêta \(p\)-adiques , Sympos. Math., Vol. XXIV (Sympos., INDAM, Rome, 1979), Academic Press, London, 1981, pp. 315-345. · Zbl 0463.14016
[17] A. Néron, Fonctions thêta \(p\)-adiques et hauteurs \(p\)-adiques , Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981), Progr. Math., vol. 22, Birkhäuser Boston, Boston, MA, 1982, pp. 149-174. · Zbl 0492.14035
[18] P. Norman, \(p\)-adic theta functions , Amer. J. Math. 107 (1985), no. 3, 617-661. JSTOR: · Zbl 0587.14028
[19] B. Perrin-Riou, Descente infinie et hauteur \(p\)-adique sur les courbes elliptiques à multiplication complexe , Invent. Math. 70 (1982/83), no. 3, 369-398. · Zbl 0547.14025
[20] B. Perrin-Riou, Sur les hauteurs \(p\)-adiques , C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 6, 291-294. · Zbl 0532.14012
[21] J. Tate, The arithmetic of elliptic curves , Invent. Math. 23 (1974), 179-206. · Zbl 0296.14018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.