Analytic subvarieties with many rational points
| dc.creator | Gasbarri, Carlo | |
| dc.date | 2008-11-19 | |
| dc.date.accessioned | 2026-07-07T10:19:33Z | |
| dc.date.available | 2026-07-07T10:19:33Z | |
| dc.description | We give a generalization of the classical Bombieri--Schneider--Lang criterion in transcendence theory. We give a local notion of $LG$--germ, which is similar to the notion of $E$-- function and Gevrey condition, and which generalize (and replace) the condition on derivatives in the theorem quoted above. Let $K\subset \Bbb C$ be a number field and $X$ a quasi--projective variety defined over $K$. Let $γ\colon M\to X$ be an holomorphic map of finite order from a parabolic Riemann surface to $X$ such that the Zariski closure of the image of it is strictly bigger then one. Suppose that for every $p\in X(K)\capγ(M)$ the formal germ of $M$ near $P$ is an $LG$-- germ, then we prove that $X(K)\capγ(M)$ is a finite set. Then we define the notion of conformally parabolic Khäler varieties; this generalize the notion of parabolic Riemann surface. We show that on these varieties we can define a value distribution theory. The complementary of a divisor on a compact Khäler manifold is conformally parabolic; in particular every quasi projective variety is. Suppose that $A$ is conformally parabolic variety of dimension $m$ over $\Bbb C$ with Khäler form $ω$ and $γ\colon A\to X$ is an holomorphic map of finite order such that the Zariski closure of the image is strictly bigger then $m$. Suppose that for every $p\in X(K)\cap γ(A)$, the image of $A$ is an $LG$--germ. then we prove that there exists a current $T$ on $A$ of bidegree $(1,1)$ such that $\int_AT\wedgeω^{m-1}$ explicitly bounded and with Lelong number bigger or equal then one on each point in $γ^{-1}(X(K))$. In particular if $A$ is affine $γ^{-1}(X(K))$ is not Zariski dense. | |
| dc.description | 44 pages. Submitted | |
| dc.identifier | https://arxiv.org/abs/0811.3195 | |
| dc.identifier | http://arxiv.org/abs/0811.3195 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/174547 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Number Theory | |
| dc.subject | 11J97; 14G40; 30D35 | |
| dc.title | Analytic subvarieties with many rational points | |
| dc.type | text |