# Questions tagged [arithmetic-geometry]

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

1,314 questions
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### How to visualize the Frobenius endomorphism?

As the question title asks for, how do others "visualize" the Frobenius endomorphism? I asked some people in real life and they said they didn't know and that I could go and ask on MO and possibly get ...
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### Mordel's conjecture for function fields in positive characteristic

Manin proves Mordel's conjecture for function fields in characteristic zero.his proof has a gap but Coleman fill this gap and restate Manin proof in a more modern language.both of them work over ...
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### An explicit correspondence for reductions of modular curves $Y(N)$

Let $Y(N)$ be the modular curve associated with the principal congruence subgroup $\Gamma(N) \subset \mathrm{SL}(2, \mathbb{Z})$ of level $N \in \mathbb{N}$. It is well known that this curve has a ...
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### Can an algebraic variety over a field $k$ be the union of proper closed subsets $(S_i)_{i\in I}$ with $I < k$

Let $k$ be an algebraically closed field (of characteristic zero, if it helps). Let $X$ be an algebraic variety over $k$. Let $I$ be an index set such that the cardinality of $I$ is smaller than the ...
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### Reference request: Newton above Hodge

Let $K$ be a p-adic field, and let $\mathcal{O}$ be the ring of integers inside $K$ with residue field $k$. Let $\mathcal{X}$ be a smooth proper formal scheme over $\mathcal{O}$ (with topology given ...
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### Singular abelian surfaces that can be defined over $\mathbb Q$

An abelian surface $A$ is called singular if it has maximal Picard number $\rho(A) = 4$. By work of Shioda-Mitani, any singular abelian surface $A$ is the product $A = E_1 \times E_2$ of two ...
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### Topological finiteness of etale fundamental group for arithmetic schemes

By comparison theorem and finite CW complex structure we know every complex variety's etale fundamental group is topological finitely generated (see here). It seems natural to ask similiar things for ...
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### Non algebraizable formal abelian schemes

I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable. If ...
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### Is the Fermat quartic surface a generalized Zariski surface?

Consider the Fermat quartic surface $$F\!: x^4 + y^4 + z^4 + t^4 = 0$$ over an algebraically closed field $k$ of odd characterstics $p$. Shioda proved that for $p=3$ this surface is a generalized ...
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### Smooth algebras always lift

Let $k$ be a finite field, $A$ a smooth $k$-algebra. Does there exists a smooth algebra $B$ over the Witt vectors $W(k)$, such that $B/p\simeq A$? How is it constructed?
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### Liftings and closed immersions

Let $A$ be a flat $\mathbf{Z}_p$-algebra, $\overline{I}\subset A/p$ an ideal. Can we find an ideal $I\subset A$ such that $I$ mod $p$ = $\overline{I}$ $I$ does not contain $p$. It's harder than it ...
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### Under what conditions are superspecial abelian surfaces isomorphic over a finite field?

Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...
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### Coherent modules over complete adic rings: counterexamples

Let $A$ be a coherent ring, complete with respect to the adic topology generated by a finitely generated ideal $I$. Define the category $Coh(A,I)$ whose objects are inverse systems $\{M_n\}$ of $A$-...
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### Absolute approximation of formal schemes

Let $\mathfrak{X}_j$ be an inverse system of qcqs $p$-adic formal scheme, flat over $\mathbf{Z}_p$, with affine transition maps, and assume $\mathcal{O}_{\mathfrak{X}_j}$ is a coherent sheaf of ...
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### Liftability of varieties, after fpqc base change

Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable. Suppose there exists an fpqc cover $S'\to S$, such ...
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### affine Lefschetz and Poincaré duality for syntomic cohomology

Let $X$ be a smooth variety over an algebraically closed field $k$ of characteristic $p > 0$. Is there an affine Lefschetz theorem and Poincaré duality for sheaves represented by finite flat ...
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### Is there any generalization of Weil conjecture for non-smooth variety?

Is there any generalization of Weil conjecture for any non-smooth geometric-connected variety? For example, for more general curve, or at least some numerical evindence (example)?
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### Do non-constant maps specialize to non-constant maps?

Let $R$ be a dvr with fraction field $K$ and residue field $k$. Let $\mathcal{X}\to \mathcal{Y}$ be a morphism of $R$-schemes such that $\mathcal{X}_K\to \mathcal{Y}_K$ is non-constant. Is the ...
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### Approximation of constructible abelian étale sheaves

Let $F$ be a constructible abelian étale sheaf of modules over a finite ring $\Lambda$ on a scheme $X$ over a field $k$, with the size of $\Lambda$ invertible on $X$. Suppose $X = \varprojlim X_j$, ...
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### Projective embeddings and quasi-compactness

Let $X$ be a projective scheme over a ring $R$, and $p : X\to\mathbf{P}^n_R$ a projective embedding. Does there exist $n$ large enough so that the complement $U\subset \mathbf{P}^n_R$ of $p(X)$ in ...
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### Descent of étale torsors

Let $X$ be a scheme over a field $k$, $G$ a finite abelian group of size invertible on $X$. Suppose $K/k$ is a Galois field extension and let $Y\to X_K$ be an étale $G$-torsor. For what field ...
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### Is there a relationship between the zeta function of a Laplacian and the Selberg Zeta Function?

Let us define the zeta function of an elliptic differential operator $H$ with eigenvalues $\lambda_n$ like so: \begin{aligned} \zeta_H(s) := tr( H^{-s} ) \\ := \sum^\infty_{n=1} ...