Multiplicative and additive relations for values of rational functions and points on elliptic curves

Abstract: For given rational functions $f_1,\ldots,f_s$ defined over a number field, Bombieri, Masser and Zannier (1999) proved that the algebraic numbers $\alpha$ for which the values $f_1(\alpha),\ldots,f_s(\alpha)$ are multiplicatively dependent are of bounded height (unless this is false for an obvious reason).

Motivated by this, we present various extensions and recent finiteness results on multiplicative relations of values of rational functions, both in zero and positive characteristics. In particular, one of our results shows that, given non-zero rational functions $f_1,\ldots,f_m,g_1,\ldots,g_n \in \mathbb{Q}(X)$ and an elliptic curve $E$ defined over $\mathbb{Q}$, for any sufficiently large prime $p$, for all but finitely many $\alpha\in\overline{\mathbb{F}}_p$, at most one of the following two can happen: $f_1(\alpha),\ldots,f_m(\alpha)$ satisfy a short multiplicative relation or the points $(g_1(\alpha),\cdot), \ldots,(g_n(\alpha),\cdot)\in E_p$ satisfy a short linear relation on the reduction $E_p$ of $E$ modulo $p$.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

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