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### Points with minimal height

Let $K$ be an algebraically number field and $$\phi : \mathbb P^n (K) \to \mathbb P^m (K)$$ a polynomial map, such that $\forall \alpha \in \mathbb P^n$, where $\alpha = [\alpha_0, \dots , \alpha_n]$, ...

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### rank of Abelian schemes under ample hypersurface section

Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface ...

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### Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR

Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of ...

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### Elliptic curve E and Galois representation

Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then
Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$?
Next
...

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### Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure

This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...

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### Absolute Hodge implies Galois invariant?

Let $X$ be an Abelian variety defined over a number field $K$, suppose that it has a good reduction over a fine place $\mathfrak{p}$ of $K$. Let $G_{\mathfrak{p}}$ be the local Galois group for ...

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### Elliptic curve and Galois representation

For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by
$\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = ...

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### vanishing of étale cohomology of affine surface

Let $U$ be an affine smooth surface over an algebraic closure of a finite field. Let $\mathscr{A}/U$ be an Abelian scheme and $\ell \neq \mathrm{char}(k)$ be prime.
Are there vanishing results for ...

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### Specialization Map of family of abelian varieties

In Lang's Survey on Diophantine Geometry, page 40, he said the following:
Let $F=k(Y)$ be a function field of variety $Y$ over the constant field $k$ and $X_F$ a non-singular projective variety over ...

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### kernel of isogeny becomes constant after base change

Let $S = Spec(O_K)$ be the spectrum of the rings of integers of a number field $K$. Let $A/S \setminus T$ be an Abelian scheme over an open subscheme $S \setminus T \subseteq S$. Does the kernel of ...

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### Local factors of Hasse-Weil zeta function - what do they have in common?

Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...

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### Bounding failures of the integral Hodge and Tate conjectures

It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...

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### Families of Fano varieties over non-hyperbolic curves

Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G_m$, or an elliptic curve.
Let $f:X\to C$ be ...

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### paper by Nakata on 2-adic Galois representations

In Keisuke Arai's 2007 paper "On uniform lower bound of the Galois images associated to elliptic curves", which can be found on ArXiv, Arai makes the following citation:
K. Nakata. On the 2-adic ...

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### Torsion group of the following elliptic curve

Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i $ be the elliptic curve defined by $y^2 = x^3 + n_i$.
Can one compute the torsion group ...

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### Does the isotropic group of local galois representation have finite index?

Let $\rho : G\to \mathrm{GL}_n(\mathbb Z_p)$ be a crystalline representation of $G=\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$. For any non zero element $a\in \mathbb{Z}_p^n$ , it spans a rank one ...

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### Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology

This question is inspired by the question: Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
Let $K$ be a number field (or finitely generated field of ...

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### log smooth curve vs pointed node curve

F.Kato has a statement said that a log smooth curve $f:(X, M,\alpha)\to (k,N,\beta)$ where $k$ is an algebraic closed field and $N$ is some fine log structure on $k$, is equvalent to pointed node ...

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### log structure on Witt ring and Frobenius map

I am a little confused about the log structure of Witt ring and its Frobenius map.
Let $k=\bar{\mathbb{F}}_p$, $W:=W(k)$ the Witt ring.
We know that to deal with the semistable reduction, i.e. scheme ...

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### Deforming curves to other curves over the field of rational numbers

Let $X$ and $Y$ be smooth projective geometrically connected curves over $k$ of genus $g$ at least two.
If $k$ is an algebraically closed field of characteristic zero, there exists a connected ...

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### Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is

Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space.
If $\mathcal X$ can be defined over ...

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### Question about the “middle” intermediate Jacobian

Suppose $Y$ is a smooth projective variety of dimension $2p-1$ over $\mathbb{C}$. I have a few questions about the $p^{~\text{ th}}$ intermediate Jacobian $J^p(Y)$ of $Y$.
Does it come from (i.e. is ...

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### Are there generating functions of rational or integral solutions of Diophantine equation that

As we know,there are generating functions for c.e.languages which are some retricted rational or algebraic or transcendental functions dependent on the class of languages like regular ...

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### representation of algebraic fundamental group of projective line minus three point

everyone, I want to ask is there any result in the literature
similar to the following:
Let $ X=\mathbb{P}^1\backslash \{0,1,\infty\}$, then $X$ is defined over $\mathbb{Z}$. Let $X_{\mathbb{Q}}$ ...

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### degree of isogenies between Jacobians and Abelian Varieties

Let $K$ be a local field of characteristic zero and positive residual characteristic. Let $A$ be a simple abelian variety and assume we have an isogeny $f:Jac_C\rightarrow A$ with $C$ a smooth curve ...

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### What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the ...

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### Existence of local sections

I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.
Take a number field $K$, and let ...

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### integral hard Lefschetz

I am looking for examples $(X,\eta)$ where the integral hard Lefschetz is an isomorphism:
$X/k$ is a smooth projective variety of dimension $d$ over an algebraic closure of a finite field and $\eta ...

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### Arithmetic property of a surface of general type

In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...

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### What is the probability that a randomly chosen number from set of c.e.number is period(number)?

What is the probability that a randomly chosen number from the set of c.e.numbers is period(number)?
What is the probability that a randomly chosen number from the set of computable numbers is ...

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### What is the relation between KC and height of rational number?

Roughly speaking,Kolmogorov Complexity of a bits string or a description is the minimal length of programs outputing a bits string,and height of rational number is logarithm of the largest numerator ...

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### Determining existence of p-adic point on a plane curve

Is there an implemented algorithm that will take a polynomial $f(x,y)\in\mathbb Z[x,y]$ and a prime $p$, and determine whether the equation $f(x,y)=0$ has a solution over $\mathbb Q_p$?

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### Etale cohomology and restricted direct product

[migrated from math.stackexchange: http://math.stackexchange.com/questions/727896/etale-cohomology-and-restricted-direct-product]
$\newcommand{\h}{\operatorname{H}}$
Let $k$ be a global field, $A$ an ...

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### Can the failure of the multiplicativity of archimedean L-factors be corrected?

My question is parallel to J. Borger' question:
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
As emphasized by Scholbach in his paper on special values of ...

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### Bounds for the Tamagawa number of the Jacobian of a hyperelliptic curve

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ and let $C$ be a hyperelliptic curve of genus $g$ defined over $K$ with Jacobian $J$.
Suppose that $C$ is given by ...

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### Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...

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### Totally real points on curves

Let $X$ be a smooth, projective (geometrically integral) curve defined over $\mathbb{Q}$ with genus $g \geq 3$. Suppose that $X(\mathbb{R}) \neq \emptyset$. Does $X$ have a point defined over a ...

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### Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)

Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in ...

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### Finite morphisms to projective space

Let $X$ be a projective variety of dimension n. Then there exists a finite surjective morphism $X \to \mathbf P^n$. Let $d$ be the minimal degree of such a finite surjective morphism.
Let $d^\prime ...

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### Degeneration of the Hodge spectral sequence for scheme over truncated Witt ring?

Let $f:X\to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb{Q}$-scheme), then Deligne has shown:
The Hodge-De Rham spectral sequence ...

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### Number field of degree 5

I am interested in field extensions of the rationals. About degree 3 extensions there are many refrences including the famous paper of Shanks "The Simplest Cubic Fields". In particular he gave ...

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### Potential good reduction of abelian varieties

In Corollary 3 on page 498 of the article "Good reduction of abelian varieties" it says that, under some specified conditions, the minimal subextension $L/K$ of $\overline{K}/K$ over which an abelian ...

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### Base change in crystalline cohomology?

Does one have a base change theorem in crystalline cohomology like in étale cohomology?
Suppose one has the following cartesian diagram
$$
...

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### What is “special” maximal compact subgroup of algebraig group over local field?

Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field.
Here, I think the word "compact" is used ...

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### Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...

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### Explicit calculation of Weil Deligne representations

According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations.
However, given a galois representation, it is usually difficult ...

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### Is there a largest prime p such that J_0(p) completely splits into elliptic curves

The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...

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### The Rappoport-Zink spectral sequence vs. the one of the complement of a normal crossing divisor

As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale ...

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### Covers of modular curves

I'm interested in covers of modular curves (especially cyclic covers) and I'm sure there's a lot of information out there available on this topic. However, I'm unable to locate any literature (on ...

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### A problem in intersection theorem

I'm reading the paper:
SGA 7 II, Intersections sur les surfaces regulieres.
In Papge 6 , I cannot understand why there is sign $-1$ in the formula (1.10.4):
Let $S$ be a trait, for any $\mathcal ...