# Questions tagged [arithmetic-geometry]

Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

1,452
questions

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82 views

### Varieties with everywhere good reduction isomorphic over every completion

Let $R$ be the ring of integers in a number field. Let $X$ and $Y$ be smooth and proper schemes over $R$. For a maximal ideal $\mathfrak{m}\subset R$ denote the localization of $R$ at $\mathfrak{m}$ ...

**3**

votes

**2**answers

65 views

### primes of multiplicative reduction for elliptic curves with rational $\ell$-torsion

Very recently, I made an observation from scanning lists of elliptic curves on the LMFDB that leads me to the following (unproven) statement:
Fix $\ell \in \{5, 7\}$. Let $E$ be an elliptic curve ...

**26**

votes

**3**answers

4k views

### Why is this “the first elliptic curve in nature”?

The LMFDB describes the elliptic curve 11a3 (or 11.a3) as "The first elliptic curve in nature". It has minimal Weierstraß equation
$$
y^2 + y = x^3 - x^2.
$$
My guess is that there is some problem ...

**0**

votes

**0**answers

66 views

### Chinese remaindering to solve solvable diophantine equations

Given a diophantine equation $$f(x_1,\dots,x_z)=0$$ where $f(x_1,\dots,x_z)\in\mathbb Z[x_1,\dots,x_z]$ is of total degree $d$ and each variable degree $d_i$ where $i\in\{1,\dots,z\}$ there is no ...

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**2**answers

86 views

### A variant of Turán–Kubilius inequality

Let $\omega(n)$ the number of distinct prime factors of $n$ (counted without multiplicity). A famous consequence of Turán–Kubilius inequality is
$$
\sum_{n\leq x}(\omega(n)-\log\log x)^2=O(x\log \log ...

**6**

votes

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282 views

### Coherent cohomology of the generic fiber of Lubin-Tate space vs. of Lubin-Tate space considered rationally?

I am trying to compare the coherent cohomology of the generic fiber of Lubin-Tate space to the coherent cohomology of Lubin-Tate space considered rationally, and I am going in circles! I would be very ...

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333 views

### Open conjectures and expected applications of homotopy theory to arithmetics

I hope this question is not too broad to be asked here; if it is, please feel free to close the question.
I'm currently near the end of my masters studies and subsequently search for a particular ...

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113 views

### A complex analytic version of the eigencurve

I am very much a beginner to the theory of eigencurves so there might be many mistakes in what follows, especially since it is all very speculative.
My understanding of the eigencurve $\mathcal C_{N,...

**6**

votes

**1**answer

245 views

### On the moduli stack of abelian varieties without polarization

(I am especially interested in abelian surfaces and characteristic 0).
How bad is the moduli stack of abelian varieties (with no polarization or level structure)? Is it an Artin stack? DM (Deligne-...

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101 views

### reduction type of elliptic curves over $p$-adic fields and local Langlands correspondence for $GL_2(F)$

In an introductory note of local Langlands correspondence http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/old_introductory_notes_on_local_langlands.pdf, section $11$ describes a recipe to ...

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115 views

### $n$-variable polynomials modulo $p$

The Hasse-Weil bound implies that for any 2-variable polynomial $P(x,y)$, there exists approximately $p$ solutions in $\mathbb{F}_p$ of $P(x,y) \equiv a \pmod p$ for sufficiently large $p$, and any ...

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63 views

### Existence of a “p-adic Mahler measure” or alternatively, the converge of a p-adic sequence

Let $f \in \mathbb Z_p[[t]]^\times$ be an invertible power series and let $\log_p$ be the p-adic logarithm with the normalization that $\log p = 0$. Consider the sequence:
$$a_n = \frac{1}{p^{n-1}}\...

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**1**answer

174 views

### perfectoid field of characteristic $p$

Let $L$ be perfectoid field of characteristic $p$ and $L'$ be a finite extension of $L$. Then I want to prove the trace map $\text{Tr}_{L'/L}: m_{L'}\rightarrow m_L$ is surjective. I find a proof in ...

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176 views

### Explicit construction of the Jacobian of a curve

Let $k$ be an algebraically closed field (of arbitrary characteristic), and $C$ a smooth projective curve over $k$, given by defining equations in projective space. I am looking for an algorithmic ...

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**1**answer

162 views

### Which $p$-adic valuations of Weil numbers (that is, eigenvalues of Frobenius) are possible?

Let $C$ be a smooth projective curve over a finite field $\mathbb F_q$, $q$ is a power of the characteristic $p$. It is well-known that if $\alpha$ is an eigenvalue of Frobenius acting on $H^1_{et}(C,\...

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73 views

### Hasse invariant of abelian varieties with complex multiplication

Is there a good way to compute Hasse invariants of elliptic curves or higher dimensional Abelian varieties with complex multiplication?
For example, if $E$ is an elliptic curve with CM by an ...

**4**

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**0**answers

89 views

### The profinite topology on the Mordell Weil group

In this lecture of Serre on his open image theorem, around 6 minutes, Serre mentions the following theorem of Tate:
Let $A/k$ be an abelian variety over a number field and consider the Mordell-Weil ...

**4**

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136 views

### Beilinson regulator: a road map

I'm approaching to the Beilinson Conjecture and after studying some properties of the Deligne-Beilinson cohomology, I want to understand the regulator maps. But I don't know anything about K-theory ...

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**1**answer

136 views

### Translates of abelian subvarieties

Suppose $A$ is an abelian variety over an algebraically closed field $k$. I'm confused about the notion of translates of abelian subvarieties. I looked over a few related papers, but I couldn't find a ...

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112 views

### A Lefschetz style formula for the $\ell^\infty$ torsion of an Abelian variety over a finite field

Let $A/\mathbb F_q$ be an abelian variety over a finite field. Define $A_\ell = A[\ell^\infty](\mathbb F_q)$, the $\ell^n$ ($n\geq 0)$ torsion points defined over the base field. I can assume $\ell \...

**7**

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**1**answer

195 views

### Finding $Q(\sqrt{-2})$-rational points on $X_0(33)$

Let $K = Q(\sqrt{-2})$. How can I compute the $K$-rational points on the modular curve $X_0(33)$?
Recall that $X_0(33)$ is of genus $3$ and has the following affine model,
$$y^2 +(-x^4-x^2-1)y = 2x^...

**7**

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**1**answer

183 views

### The $S$-unit equation for functions on curves

Let $X$ be a smooth projective connected curve over a number field $k$, and let $S \neq \emptyset$ be a finite set of closed points of $X$. The curve $Y = X \setminus S$ is affine, and we denote by $R$...

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**1**answer

109 views

### Clarification: Using Hensel's Lemma to determine $K_v$-rational points on a curve

From Silverman's AEC page 332:
I need to understand why the determination of the following local kernel
$$
ker \Big( H^1(G_v, E[\phi]) \rightarrow WC(E/K_v)[\phi] \Big)
$$
is straightforward. The ...

**4**

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153 views

### Is having no rational point always witnessed over a place?

Let $K$ be a finitely generated extension of $\mathbb{Q}$ of transcendence degree at least 1.
Recall that a valuation ring of $K/\mathbb{Q}$ is a sub-$\mathbb{Q}$-algebra $V\subset K$ such that for ...

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88 views

### galois deformation ring with type is union of irreducible components

Notation:
$K$ finite extension of $\mathbb{Q}_p$, $G_K$ absolute Galois group of $K$,
$E$ is finite extension of $\mathbb{Q}_p$ (coefficient field), $O_E$ is ring of integer in $E$.
In this paper of ...

**4**

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120 views

### Kottwitz global gerbes

I've been trying to understand global gerbes constructed by Kottwitz in $B(G)$ for all local and global fields (arXiv:1401.5728). Scholze explained in $p$-adic geometry (arXiv:1712.03708) why I would ...

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98 views

### Group cohomology of sheaves under closed immersion

Suppose $X$ is a scheme over Spec $\mathbb{Z}$, and $p$ is a non-zero prime in $\mathbb{Z}$. Then we have a closed immersion from the special fibre $i_p: X_p \rightarrow X$. If $\mathscr{F}$ is a ...

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81 views

### Inseparable field extensions of degree p and linear independence

Let $F$ be a field of characteristic $p$; let $\alpha \in F$ such that $\alpha \neq \beta^p$ for any $\beta \in F$, and let $K := F(x)$ where $x=\sqrt[p]{\alpha}$.
Is it true that the elements $1,(x-...

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98 views

### L-function in p-adic spaces

I've been learning more about different $p$-adic geometries, namely Berkovich spaces, Huber's Adic spaces and ridgid analytic spaces. In arithmetic geometry, it is often very interesting to assoicate ...

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157 views

### Existence of polynomials $p_n(x,y)$ such that FLT is true for $n\geq 3$

Since $x^n+y^n=z^n$ has no solutions in integers for $n \geq 3$ I started to think about polynomials of degree $\leq n-1$ which need to be added to $x^n+y^n$ so that $x^n+y^n+p_n(x,y)=z^n$ has an ...

**4**

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**1**answer

189 views

### Weight 3 modular form associated to singular abelian surfaces?

Given an extremal K3 surface $S$ over $\mathbb{Q}$ (i.e. a K3 surface with maximal Picard rank) there is a 2-dimensional Galois representation on the transcendental lattice $T(S)$, and an associated ...

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122 views

### Does the cardinality of coordinate projections of the rational points of affine varieties over finite fields also tend to $\infty$?

We know (basically by Lang-Weil) that for an absolutely irreducible n-dimensional affine variety $V$ the cardinality $\#V(F_{l})$ tends to $\infty$ for $l$ large enough. We could now look at the set ...

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152 views

### No intermediate denominators growth for holonomic functions?

My question concerns holonomic sequences of rational numbers, meaning assignments $a(n) : \mathbb{N} \to \mathbb{Q}$ fulfilling a linear recurrent relation of the form
$$
a(n+k) = \sum_{i=0}^{k-1} p_i(...

**1**

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**1**answer

95 views

### Elliptic curves and its Neron model

Let $E$ be an elliptic curve over $\mathbb{Q}$. For a prime $p$, let $\mathcal{E}_p$ denote its Neron model over $\mathbb{Z}_p$. Also, let $\Phi_p(E)$ denote the component group of $\mathcal{E}_p$.
...

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86 views

### Some questions regarding computation of the Mordell-Weil group

I was reading the theory relevant to Selmer and Shafarevich-Tate groups from Silverman's AEC. And I have a lot of doubts related to these topics:
First, I don't understand the reasoning behind the ...

**7**

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260 views

### Stacky proof of no elliptic curves over Z

It is a well known result that there are no Elliptic curves over the integers with every where good reduction. In fact this is even true for abelian varieties (and hence higher genus curves) but let ...

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162 views

### Finiteness results in the category of schemes up to $\mathbb{A}^1$-homotopy

In algebraic geometry, we know that there exist geometrical conditions on a scheme $X/k$ for having finitely many rational points when $k$ is a number field. Namely for curves there is the Mordell ...

**22**

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**1**answer

551 views

### Geometry of algebraic curve determined by point counts over all number fields?

Let $C$ be a smooth (geometrically irreducible) projective curve of genus $g>1$ over a number field $K$. The Mordell conjecture (first proved by Faltings) says that for any finite field extension $...

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205 views

### Equivalent definitions of the ring $B_{\mathrm{cris}}$

I'm reading Laurie's note about Fargues-Fontaine Curve and I think he uses a different definition of $B_{\mathrm{cris}}$. Usually when $R$ is a perfect ring of characteristic $p$, $A_{\mathrm{cris}}(R)...

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150 views

### Good reduction of finite etale covers of abelian varieties

Let $R$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $K$.
Let $A$ be an abelian variety over $K$ with good reduction over $R$. Let $X\to A$ be a finite etale ...

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94 views

### Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)?

I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist ...

**4**

votes

**1**answer

280 views

### Example of a non-odd motive appearing in cohomology of intermediate degree

I would like to know an example of a projective variety over a totally real field where a complex conjugation is not odd on some of its étale cohomology.
Edit: I am looking for the most interesting ...

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56 views

### Over derivations with an inusual property

Let $A$ be a ring finitely generated over $\mathbb{Z}$. Let $D$ be a derivation on $A$. Let $D^p$ be the composition of $D$ with itself, $p$ times. We suppose that $D^p(x)$ belongs to the ideal $pA$, ...

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127 views

### Divisors in number field / function field analogy

My question concerns a specific entry on Bjorn Poonen's table. He states that the correct analogue of the divisor group of a smooth projective curve over $\mathbb{F}_q$ is the group of Arakelov ...

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171 views

### Independence of $p$ of Hodge-Tate weights

Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $R_p=H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\...

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86 views

### Finite locally free group scheme killed by its order?

When I saw this paper of René Schoof, there are two questions on the first page and what confuses me is that how to reduce the first question to the second. This is expalined in the first paragraph on ...

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**1**answer

315 views

### Irreducible global Galois representation with weights 0, 1, 3?

Fix a prime number $p$. Can there exist a continuous irreducible representation $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{GL}_3(\mathbb{Q}_p)$ that is unramified at almost all primes, ...

**4**

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**2**answers

243 views

### The diagonal of the Weil restriction

Let $Y\to X$ be a finite surjective morphism of smooth projective geometrically connected varieties over $\mathbb{Q}$. Let $k$ be a number field and consider the induced morphism
$$f:Res_{k/\mathbb{...

**6**

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**1**answer

157 views

### Functional equation of twisted triple product L-function

Let $\mathbb{E}=E_1\times E_2\times E_3$ denote the product of three elliptic curves over $\mathbb{Q}$ of prime level $p$ and consider the $p$-adic Galois representation $$V_p(\mathbb{E})=H^1_{et}(E_{...

**3**

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**1**answer

113 views

### How does an analytic space correspond to a $p$-adic Banach space

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $V$ be a Banach algebra over $K$, then what is the $K$-analytic space corresponding to $V$? What is the definition of $K$-analytic space? This is ...