# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

1,314 questions
0answers
58 views

### Weil cohomology theories “genuinely” of positive characteristic

One of the reasons why Weil cohomology theories are required to have coefficients in a field of characteristic 0 is that they are supposed to be robust enough to solve Weil conjectures, i.e. to count ...
0answers
213 views

0answers
108 views

0answers
100 views

### Unique way to topologise finite algebra over Huber ring

Let me start with the following Lemma. $\textbf{Lemma}$ Let $A$ be a Tate ring, and let $f\colon A\to B$ be a finite $A$-algebra. Then there is a unique way to topologise $B$ turning it into a ...
0answers
65 views

### Current status of uniform boundness of rational points on higher genus curves

We know Lang conjecture can imply uniform boundless of rational points on higher genus (smooth projective) curves over a fixed number field by works of Mazur and others. How is the conjecture of ...
0answers
88 views

1answer
191 views

1answer
417 views

### Smooth proper variety over $\mathbb Q$ with everywhere bad reduction

$\newcommand{\Spec}{\operatorname{Spec}}$ Cross-post from Math.SE, hopefully people more knowledgeable in the field will see the question here on MO. It is a well-known fact that a smooth projective ...
0answers
133 views

### Berthelot’s comparison theorem and functoriality

Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$. Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...