Quasi-equivalence of Heights and Runge's Theorem

Let $P$ be a polynomial that depends on two variables $X$ and $Y$ and has algebraic coefficients. If $x$ and $y$ are algebraic numbers with $P(x,y)=0$, then by work of N\'eron $h(x)/q$ is asymptotically equal to $h(y)/p$ where $p$ and $q$ are the partial degrees of $P$ in $X$ and $Y$, respectively. In this paper we compute a completely explicit bound for $|h(x)/q-h(y)/p|$ in terms of $P$ which grows asymptotically as $\max\{h(x),h(y)\}^{1/2}$. We apply this bound to obtain a simple version of Runge's Theorem on the integral solutions of certain polynomial equations

The Tate-Voloch Conjecture in a Power of a Modular Curve

Let $p$ be a prime. Tate and Voloch proved that a point of finite order in the algebraic torus cannot be $p$-adically too close to a fixed subvariety without lying on it. The current work is motivated by the analogy between torsion points on semi-abelian varieties and special or CM points on Shimura varieties. We prove the analog of Tate and Voloch's result in a power of the modular curve Y(1) on replacing torsion points by points corresponding to a product of elliptic curves with complex multiplication and ordinary reduction. Moreover, we show that the assumption on ordinary reduction is necessary.Comment: Corrected some typos in version

Bad reduction of genus 22 curves with CM jacobian varieties

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a formula by Colmez and Obus specific to the CM case and valid when the CM field is an abelian extension of the rationals. This formula links the height and the logarithmic derivatives of an $L$-function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use results of Igusa, Liu, and Saito to show that the contribution at the finite places in our decomposition measures the stable bad reduction of the curve and subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang to handle the infinite places

A Note on Divisible Points of Curves

Let $C$ be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3. We partially answer a question posed by Levin on points on $C$ for which a non-trivial power lies again on $C$. Our results have connections to Zilber's Conjecture on Intersections with Tori and yield to methods arising in transcendence theory and the theory of o-minimal structures.Comment: Published version, but with an error fixed in the formula for the function on page