Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
1,452
questions
1
vote
0answers
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
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 ...
26
votes
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 ...
0
votes
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 ...
0
votes
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
votes
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 ...
10
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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 ...
4
votes
0answers
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
vote
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 ...
8
votes
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
votes
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,\...
2
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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,
$$y^2 +(-x^4-x^2-1)y = 2x^...
7
votes
1answer
183 views
The $S$-unit equation for functions on curves
Let $X$ be a smooth projective connected curve over a number field $k$, and let $S \neq \emptyset$ be a finite set of closed points of $X$. The curve $Y = X \setminus S$ is affine, and we denote by $R$...
1
vote
1answer
109 views
Clarification: Using Hensel's Lemma to determine $K_v$-rational points on a curve
From Silverman's AEC page 332:
I need to understand why the determination of the following local kernel
$$
ker \Big( H^1(G_v, E[\phi]) \rightarrow WC(E/K_v)[\phi] \Big)
$$
is straightforward. The ...
4
votes
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
vote
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
votes
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 ...
1
vote
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
votes
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
vote
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 ...
0
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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 ...
