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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.

MSC:
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
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