1
vote
1answer
123 views
Why is the base change functor faithful
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 …
5
votes
1answer
193 views
Potentially good, semi-stable reduction => good reduction ?
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 …
6
votes
1answer
516 views
Is Gouvêa-Mazur’s “Infinite Fern” a fractal?
[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 …
2
votes
4answers
572 views
Roadmap to reach Arithmetic Geometry for a Physics Major
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 …
3
votes
1answer
90 views
Rational points on the curve y^p=f(x) in characteristic p
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 …
0
votes
0answers
73 views
Density of integral (rational) points on affine varieties
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 …
8
votes
1answer
190 views
Belyi’s theorem for function fields
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 …
12
votes
1answer
386 views
Why is Faltings' “almost purity theorem” a purity theorem?
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 …
5
votes
2answers
304 views
Examples of (Phi,Gamma)-modules
What is the (Phi,Gamma)-module of an elliptic curve over Z_p, expressed by a direct construction ?
1
vote
1answer
192 views
Can the Albanese map be anything?
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 …
2
votes
1answer
86 views
Specialization of sections in an elliptic fibration
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 …
1
vote
0answers
79 views
What does Hodge theory tell us about simply connected surfaces of general type
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 …
4
votes
2answers
153 views
Can one bound the Quadratic Points on Curves?
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 …
3
votes
1answer
118 views
pro-$\ell$ etale fundamental group of a semi-abelian variety
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 …
8
votes
4answers
507 views
Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)
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 …

