Interpolation Formulas and Auxiliary Functions

✍ Scribed by Damien Roy

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 279 KB
• Volume: 94
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2595

✦ Synopsis

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.