Questions tagged [arithmetic-geometry]

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

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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 $\mathfrak{m}$ ...
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2answers
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 ...
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3answers
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 ...
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0answers
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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2answers
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
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0answers
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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0answers
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 ...
5
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0answers
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
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1answer
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-...
5
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0answers
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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0answers
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}}\...
1
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1answer
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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0answers
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 ...
4
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1answer
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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0answers
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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0answers
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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0answers
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 ...
3
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1answer
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 ...
2
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0answers
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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1answer
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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1answer
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 $R$...
1
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1answer
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 ...
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0answers
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 ...
1
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0answers
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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0answers
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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0answers
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 ...
2
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0answers
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-...
1
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0answers
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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0answers
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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1answer
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 ...
5
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0answers
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 ...
7
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0answers
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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1answer
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 $\mathcal{E}_p$. ...
2
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0answers
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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0answers
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 ...
3
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0answers
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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1answer
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 $...
4
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0answers
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)...
9
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0answers
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 ...
4
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0answers
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
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1answer
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 ...
1
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0answers
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 $pA$, ...
3
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0answers
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 ...
4
votes
0answers
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{\...
4
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0answers
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 ...
7
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1answer
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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2answers
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
votes
1answer
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
votes
1answer
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 ...

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