# Questions tagged [arithmetic-geometry]

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

1,314 questions
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### Explicit elements of the first cohomology of modular curves

Let $M$ be a modular curve and $\pi:E\to M$ the universal elliptic curve. For a prime $\ell$, let $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$. I am wondering whether there are any explicit ...
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### index of smooth varieties

What are the simplest examples of smooth, projective varieties defined over the fraction field of an Henselian DVR of characteristic $0$ which have index $>1$? EDIT: Also assume that the residue ...
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### Frobenius actions on de Rham cohomology, clarify questions on a paper of Kedlaya

I am reading the paper "$p$-adic cohomology: from theory to practice" by K. S. Kedlaya. I have several naive questions about section 2: Frobenius action on de Rham cohomology. As a physicist, I lack ...
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### Reference request: Formal Existence for stacks

Is there a formal existence Theorem for coherent sheaves on locally Noetherian Artin Stacks, in the spirit of Grothendieck's Formal GAGA? Is it available for more general stacks?
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### Integral morphism between universally closed and separated schemes

Let $f : X\to Y$ be a morphism between schemes over $\text{Spec}(\mathbf{Z}_p)$. Assume: $f$ is integral both $X$ and $Y$ are universally closed and separated over $\mathbf{Z}_p$ $f$ mod $p^n$ is an ...
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### Inverse limit of finite flat morphisms

Suppose that an inverse limit of finite flat morphisms $X_k\to S$ of qcqs schemes, with affine transition maps, is $X\to S$, such that a closed fiber of $X\to S$ is finite. Is $X\to S$ finite?
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### Relative approximation of morphisms

Let $S$ a qcqs scheme, and let $f : X := \varprojlim_j X_j \to S$ be an inverse limit of qcqs schemes $f_j : X_j \to S$ with affine transition maps. Suppose $f$ is (P). Is $f_j$ also (P) for all $j$ ...
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### Commutative algebra counterexample

Let $M$ be an $R[x]$-module, such that $M$ is finitely generated as an $R$-module. Does there exist one such $M$, such that $M\otimes_{R[x]}R[x,x^{-1}]$ is not finitely generated as an $R$-module?
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### Some questions on the $p$-adic properties of special $L$-values

Warning: Some naive, speculative questions from a total non expert. Let $\rho$ be a p-adic representation of the Galois group $Gal(\overline K/K)$ for a number field $K$. We can consider the Artin L-...
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### Arithmetic motivations for modularity in higher rank

The classical setting of modularity is that one can associate elliptic modular forms (or automorphic representations of GL(2)/$\mathbb Q$) to elliptic curves over $\mathbb Q$. This has far-reaching ...
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### Gluing finitely presented quasi coherent sheaves

Let $X$ be a quasi-compact, separated scheme, and $\{\text{Spec}(A_i)\subset X\}_{i=1,\ldots, n}$ a finite affine open cover. Suppose a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ is such ...
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### How is the Eichler-Shimura congruence related to L-functions?

My understanding is that the Eichler-Shimura relation expresses the Hecke operator $T_p$ in terms of the geometric Frobenius map. Specifically, $T_p = Frob + Ver$ for Frobenius map $Frob$ and it's ...
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### Proper mapping theorem

Let $Z\to X$ be a closed immersion of schemes. Assume $\mathcal{O}_Z$ and $\mathcal{O}_X$ both are coherent sheaves of $\mathcal{O}_Z$, resp. $\mathcal{O}_X$-modules. In particular, the coherent ...
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### A question on direct limits of rings, and descent of ideals

Motivated by an étale cohomology calculation I am going to do, here is a question that should have positive and not too hard answer. Let $A$ be a ring, $A_0$ a ring such that $A_0$ is equipped with ...
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### Galois action on functions on an adelic coset space

For a reductive group $G$ over a number field $K$, an automorphic representation for $(G, K)$ is an irreducible admissible constituent of the right regular representation of $G(\mathbb{A}_K)$ in the ...
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### Galois representations associated to the modular tower and automorphy

Consider $\mathcal{M} = \{M_{g,n}, \mu_{g,n}^{g’,n’}\}$, the system of moduli spaces of $n$-pointed smooth algebraic curves along with the basic maps amongst them coming from identifying and ...
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### Frobenius stratification of imperfect fields

Suppose $k$ is a (non-perfect) field of characteristic $p$, and $Fr$, its Frobenius map. I’d appreciate comments and references on the structure of the fitration/stratification of $k$ given by the ...
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### What are zeta functions good for?

I know a couple of answers to the above question: They can be used for point counting over finite fields/estimating the distribution of primes in characteristic 0. There are various conjectures/...
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### Geometric Frobenius on $\ell$-adic cohomology

Let $X$ be a smooth projective variety over a finite field $k$, and $F$ the geometric Frobenius on $H^*_{et}(X_{k^{sep}}, \mathbf{Z}_{\ell})$. Is $F$ only rationally bijective, or integrally ...
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### On what varieties are the conjectures on $L$-functions true

In Peter Schneider's paper "Introduction to the Beilinson conjectures", he lists some hypothesis (conjectures) on the $L$-function associated to the pure motive $H^i(X)$ where $X$ is a smooth ...
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### Smoothness in formal GAGA

Suppose $X$ is a noetherian scheme over a complete noetherian local ring $(A,\mathfrak{m})$, and assume $X$ is projective. If $\widehat{X}$, the $\mathfrak{m}$-adic formal completion, is smooth in ...
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### “Elementary” Proof that the divisor class group of varieties over finite fields is finite

Let $X$ be a geometrically integral (or geometrically reduced and geometrically connected) proper scheme over a finite field $k = \mathbb{F}_q$, so its Picard scheme exists and $\mathrm{Pic}^0_{X/k}$ ...
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### Reconstitution from reduction and tropicalization for $p$-adic varieties

For a variety $X$ over a $p$-adic field $k$, its reduction to a variety $X_p$ over residue field $k_p$ of $k$ and its tropicalization to $X_{tr}$ seem to me to be somewhat orthogonal processes. Taken ...
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### Tropicalization of perfectoid spaces and their tilts

Does tropicalization exist in the world of perfectoid spaces? Since it does for Huber's adic spaces, I thought it might for perfectoid spaces too, yet I can't find any explicit references so far. ...
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### $p$-adic lifts of tropical varieties

What is currently known about lifts of tropical varieties to varieties over $\mathbb{Q}_p$ or its extensions? Starting with an appropriate rational polyhedral cone complex what are the obstructions ...
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### Liftable rational varieties

Is there an example of a rational smooth projective variety over a perfect field of characteristic $p$, that is not liftable to characteristic zero?
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### $p$-adic lifts of varieties over finite fields

Fix a finite field $k_p$ of characteristic $p$, as well as a $p$-adic lift of it, $k_0$. In other words $k_0$ is a $p$-adic field and $k_p$ its residue field. Let $X_p$ be a non-singular variety over ...
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### Philosophical question on the role of motivic cohomology

As a physicist who has read almost nothing on the beautiful theories of motives and motivic cohomology, I have some philosophical questions on motivic cohomology. I apologize first for my naive ...
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### Ring of Witt vectors and Fontaine's deRham period ring

The construction I am interested in is the following: Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with algebraically closed residue field $k$. Let $K=Frac(V)$ and ...
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### What pure motives have Hodge realisations isomorphic to $\mathbb{Q}(0)$

Suppose $k$ is a number field and $\sigma:k \hookrightarrow \mathbb{C}$ is an embedding. If $\sim$ is an adequate equivalence relation, we can construct the category of pure motives with rational ...
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### Galois action on posets of number fields and $p$-adic fields

In the theory of group actions on posets one studies the action of a group $G$ on a poset $\mathcal{P}$ (via order-preserving permutations) partly through its derived action on the order complex of ...
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### Misunderstanding of Hodge conjecture

I ask a question on math stack exchange about Hodge conjecture and have not got any reply or comment. https://math.stackexchange.com/questions/2704988/clarify-hodge-conjecture so I decide to post ...
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In Nekovar's introductory paper "Beilinson's Conjecture" http://math.stanford.edu/~conrad/BSDseminar/refs/BeilinsonintroII.pdf The conjecture is formulated for smooth projective varieties over $\... 1answer 394 views ### Arithmetic representation stability and Galois action I am thinking about representation stability phenomena (as considered by Church, Farb, Ellenberg and others) in arithmetic settings where Galois action enters the picture, and am curious about their ... 0answers 107 views ### Characterization of a meromorphic function as arithmetic zeta function I'd like to know if there is a (conjectural) criterion for a meromorphic function on$\mathbb{C}$to be the zeta function of an arithmetic scheme, i.e., a statement of the form "If a meromorphic ... 0answers 160 views ### Geometric fundamental group and algebraically closed residue field my questions relates to the following talk of Tsuji: https://www.youtube.com/watch?v=2brDj26phP0 At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ... 1answer 161 views ### Alfred van der Poorten--rational functions paper Does anybody has a copy of the following paper: Alfred van der Poorten, Some facts that should be better known, especially about rational functions; Number Theory and Applications”, Richard A. Mollin ... 0answers 172 views ###$L$-function of induced Galois representation Suppose$L \subset K$are number fields and let's choose and fix an algebraic closure$\overline{L}$of$L$such that$L \subset K \subset \overline{L}$, hence$H:=\text{Gal}(\overline{L}/K)$is a ... 0answers 268 views ### quasi-finite group schemes and associated Galois modules Let$p$be an odd prime. Let$A$be an abelian variety over$\mathbb{Q}$and suppose that it has semistable reduction at$p$. Let$\mathcal{A}$denote the Neron model of$A$over$\mathbb{Z}_p$. Now ... 0answers 154 views ### Set theoretic complete intersections in toric varieties Is it expected that every smooth projective variety over the complex numbers, is a set-theoretic complete intersection into a smooth projective toric variety? Is there an example of a smooth ... 1answer 388 views ### Complete intersections in toric varieties Let$X$be a smooth projective variety over the complex numbers. Is$X$a global complete intersection inside a smooth projective toric variety? 0answers 155 views ### Algebraization of open formal subschemes Let$\mathfrak{X}$be a locally noetherian adic formal scheme over$\text{Spf}(A)$, with$A$an$I$-adically complete and separated noetherian ring. Suppose the mod$I$-fiber of$\mathfrak{X}$is an ... 0answers 188 views ### Neron Severi under specialization Let$X$be a smooth projective variety over$\mathbf{Q}$, and$\mathcal{X}$a smooth projective model over$\mathbf{Z}[1/N]$for$N$large enough. Call$\eta$the generic point$\text{Spec}(\mathbf{Q}...
I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem: [1] Solving genus zero diophantine ...
Let $X$ be a smooth projective variety over a field $k$ finitely generated over its prime field, and $M_{hom}(X)$ the category of motives modulo $\ell$-adic homological equivalence. (1) Is \$M_{hom}(...