Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the birational section conjecture with strong birationality assumptions
81
0
0.0
(
0
)
Authors
Giulio Bresciani
Ask ChatGPT about the research
No Arabic abstract
Let $X$ be a hyperbolic curve over a field $k$ finitely generated over $mathbb{Q}$. A Galois section $s$ of $pi_{1}(X)tomathrm{Gal}(bar{k}/k)$ is birational if it lifts to a section of $mathrm{Gal}(overline{k(X)}/k(X))tomathrm{Gal}(bar{k}/k)$. Grothendiecks section conjecture predicts that every Galois section of $pi_{1}(X)$ is either geometric or cuspidal, while the birational section conjecture predicts the same for birational Galois sections. Let $t$ be an indeterminate. We prove that, if $s$ is a Galois section such that the base change $s_{k(t)}$ to $k(t)$ is birational, then $s$ is geometric or cuspidal. As a consequence we prove that the section conjecture is equivalent to Esnault and Hais cuspidalization conjecture, which states that Galois sections of hyperbolic curves over fields finitely generated over $mathbb{Q}$ are birational.
rate research
Read More
We prove the $p$-curvature conjecture for rank two vector bundles with connection on generic curves, by combining deformation techniques for families of varieties and topological arguments.
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin-Selberg $L$-function, then the Bloch-Kato Selmer group is of rank one.
The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a Shimura variety is dense in the central leaf containing it. In this paper, we prove the conjecture for certain irreducible components of Newton strata in Shimura varieties of PEL type A and C, when $p$ is an unramified prime of good reduction. Our approach generalizes Chai and Oorts method for Siegel modular varieties.
We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $phi$ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(bP^1)^g$ has only finite intersection with any curve contained in $(bP^1)^g$. We also show that our result holds for indecomposable polynomials $phi$ with coefficients in $bC$. Our proof uses results from $p$-adic dynamics together with an integrality argument. The extension to polynomials defined over $bC$ uses the method of specializations coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of $(phi,phi)$ on $bA^2$.
In this paper, we prove the Uniform Mordell-Lang Conjecture for subvarieties in abelian varieties. As a byproduct, we prove the Uniform Bogomolov Conjecture for subvarieties in abelian varieties.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
