Let $L/k$ be a field extension of algebraically closed fields of characteristic zero. Let $U$ be a smooth quasi-projective variety over $k$. I am trying to understand why the base …

Does a smooth proper variety having semi-stable reduction as well as potentially good reduction have good reduction ? Note that over a $p$-adic field, this is true for the Galois …

[EDIT]: Following Qiaochu Yuan's comment, it is better to clarify that I do not know what the right definition of a fractal in the following question should be. But a nice answer m …

Hi Everybody! I am physics major but I read mathematics for myself. my main fields of interest are number theory and geometry. it seems that due to the works of A.Grothendieck, alg …

Let $K$ be a finite extension of $\mathbb{F}_q(t)$ and define the curve $C$ by the equation $y^p=f(x)$ where $p=\mathbf{char} K$ and $f\in K[x]$. What is the genus of $C$? When doe …

In a series of papers due to Browning, Heath-Brown, and Salberger (some of which are joint work, some individual), they established that for any projective variety $X \subset \math …

Belyi's theorem states that every smooth projective algebraic curve $C$ defined over $\bar{\mathbb{Q}}$ admits a map $C\to\mathbb{P}^1$ ramified only over $0,1,\infty$. Is there an …

My understanding of purity theorems is that they come in several flavors: 1) Those of the form "this Galois representation is pure, i.e. the eigenvalues of $Frob_p$ are algebraic …

What is the (Phi,Gamma)-module of an elliptic curve over Z_p, expressed by a direct construction ?

Sorry for the vague title. This question is about the Albanese map from the variety $M$ of canonically polarized varieties to the set of abelian varieties. (The variety $M$ is not …

Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice). Let $\eta$ be the generic po …

Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This f …

Let $C$ be a nonsingular projective curve defined over $\mathbb{Q}$, which does not admit a map of degree 1 or 2 to $\mathbb{P}^1$ or to an elliptic curve. It is then a consequence …

Let $A$ be a semi-abelian variety over $K$, $\ell$ a prime number which is not equal to char($K$). Does the abelianization of geometrically pro-$\ell$ etale fundamental group $(\p …

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$. An arithmetic subgroup is called congruence if it contains a subgroup of typ …