# Questions tagged [arithmetic-geometry]

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

1,314 questions
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### The $\mathbb{Q}$-rational cuspidal group of $J_0(N)$

Let $N$ be a positive integer and consider the modular curve $X_0(N)$ over $\mathbb{Q}$. Also, consider the Jacobian variety $J_0(N)$ of $X_0(N)$, which is an abelian variety defined over $\mathbb{Q}$....
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### Non-cuspidal Hecke eigenforms and Eisenstein series

It's a direct check that $E_{2k}(z )=\frac{\zeta(1-2k)}{2}+\sum _{n=1}^{\infty }\sigma _{2k-1}(n)q^{n}$ is an eigenform for every Hecke operator $T_n$ with eigenvalue $\sigma_{2k-1}(n)$...
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### Deligne's theorem on finite flat group schemes and generalizations

Recall Deligne's theorem that for a finite flat commutative group scheme $G$ of order $n$, the multiplication by $n$ map $[n]: G \to G$ is the zero map. I have seen the proof a few times but I can't ...
1answer
144 views

### Classes of hyperplane sections in cohomology

Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$. Is it true that the vector space $H^{2k}(X,\mathbf{Q}_{\ell}(k))^G$ has always positive ...
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### Ext group for commutative finite group schemes

EDIT: all group schemes are commutative. Thanks @R. van Dobben de Bruyn! Could anyone provide a reference request about extensions of finite group schemes / Ext groups. As far as I know the category ...
0answers
333 views

### Lifting automorphic Galois representations to arithmetic fundamental groups and their quotients

Suppose $V$ is an algebraic variety over a number field $K$. The absolute Galois group $G_K$ of $K$ acts by outer automorphisms on the étale fundamental group $\pi_1(V_{\bar{K}})$ where $\bar{K}$ is ...
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### Categorical representations of absolute Galois groups

I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.
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### Automorphy of families of motives

I have a couple of elementary questions regarding automorphy of Galois representations arising from geometric families. Suppose we have an algebraic family of varieties over a number field, and ...
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### Visualization of hidden structures in numbers

[Please allow me a note: The way desribed below allows to depict functions $f:X^2 \rightarrow Y$ completely in two dimensions (without hiding or omitting any information). This allows for depicting ...
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### Extension of Galois representations from geometry

Let $K$ be a number field. Take two representations $A$, $B$ of $G_K=\mathrm{Gal}(\bar{K}/K)$ over some ring $\Lambda$ (in fact, I only consider the case $\Lambda$ is a field, usually of ...
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101 views