# Interpolation Formulas and Auxiliary Functions

- Authors
- Journal
- Journal of Number Theory 0022-314X
- Publisher
- Elsevier
- Publication Date
- Volume
- 94
- Issue
- 2
- Identifiers
- DOI: 10.1006/jnth.2000.2595
- Disciplines

## Abstract

Abstract We prove an interpolation formula for “semi-cartesian products” and use it to study several constructions of auxiliary functions. We get in this way a criterion for the values of the exponential map of an elliptic curve E defined over Q. It reduces the analogue of Schanuel's conjecture for the elliptic logarithms of E to a statement of the form of a criterion of algebraic independence. We also consider a construction of auxiliary function related to the four exponentials conjecture and show that it is essentially optimal. For analytic functions vanishing on a semi-cartesian product, we get a version of the Schwarz lemma in which the exponent involves a condition of distribution reminiscent of the so-called technical hypotheses in algebraic independence. We show by two examples that such a condition is unavoidable.

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