Questions tagged [arithmetic-geometry]
Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.
1
vote
0answers
32 views
Algebraic theta-functions of level $2$ on an elliptic curve
Consider an elliptic curve $E\!: y^2 = x(x-1)(x-\lambda)$, where $\lambda \in k \setminus \{0, 1\}$ for some field $k$ of $\mathrm{char}(k) \neq 2$. Also consider the ample divisor $D = 2P_0$, where $...
5
votes
0answers
114 views
Fargues's Theorem for $Spa(C,C^+)$ (rather than $Spa(C,O_C)$
$\DeclareMathOperator\Spa{Spa}$Fargues's Theorem for $\Spa(C,O_C)$ states that the category of (mixed characteristic) shtukas with one paw at $x_C$ is equivalent to the category of Breuil-Kisin-...
3
votes
0answers
57 views
Jacobians of pointed curves
Let $Y$ be an algebraic curve of genus $g \geq 1$ defined over a number field $K$. If $Y$ has a $K$-point, then one can define the Abel-Jacobi map which embeds $Y$ into its Jacobian variety $\text{Jac}...
13
votes
0answers
243 views
Hensel lemma and rational points in complete noetherian local ring
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
6
votes
0answers
138 views
A refinement of Faltings' lemma
In his proof of the Mordell conjecture, Faltings proved the following important result:
Let $K$ be a number field and $S$ a finite set of primes in $K$. Then for any $g \geq 2$ there exists a number $...
4
votes
1answer
168 views
Henselianizations over countable index sets
Let $A$ be a ring, $I\subset A$ a finitely generated ideal.
The henselianization $A^h$ of $A$ along $I$ is the universal $A$-algebra that is henselian along $I$ and can be presented as a direct limit ...
1
vote
0answers
111 views
A family of crystalline representations
Let $K$ be a number field and let $v$ be a finite place of $K$. Further, let $g \geq 1$ be a positive integer. Consider the family $F(K,v,g)$ consisting of abelian varieties $A$ of dimension $2g$, ...
17
votes
1answer
309 views
Curves over number fields with everywhere good reduction
My question is the following:$\newcommand{\Q}{\Bbb Q}
\newcommand{\Z}{\Bbb Z}$
What is known about number fields $K$ fulfilling the condition
$C_{g,K}$ "there is a smooth projective curve of ...
5
votes
0answers
75 views
Relation between Faltings height and height on moduli space
Let $E$ be an elliptic curve over a number field $K$. The difference between the semistable Faltings height $h_F(E)$ of $E$ and the height $h(j_E)$ of the $j$-invariant of $E$ can be bounded in terms ...
5
votes
0answers
323 views
Is it true that all smooth group schemes can be deformed?
Consider for instance the map $\mathbb Z/p^2 \to \mathbb Z/p$ and suppose we are given a group smooth scheme $G$ over $\mathbb Z/p$. Is it always possible to lift it to a smooth group scheme $G'$ over ...
2
votes
1answer
119 views
How likely is it for Selmer groups to have mu invariant 0?
Given a number field $K$, how likely is it that we'll find at least one elliptic curve $E/K$ such that the $\mu$-invariant of its Selmer group is 0 (in a cyclotomic extension)?
7
votes
0answers
231 views
Proving infinitely many primes using algebraic geometry ideas
There are at least two well known proofs of the infinitude of primes (Euclid's original one and Euler's proof using L-series) and both of them can be extended to prove more general statements of the ...
9
votes
1answer
337 views
Nontrivial p-divisible groups over $\mathbb Z$ for general prime $p$
In Tate's famous paper about $p$-divisible groups, for a prime number $p$ he asks whether there exists a $p$-divisible group $G$ over $\mathbb Z$ such that $G$ is not a direct sum of $\mu_{p^\infty}$ ...
3
votes
0answers
143 views
Tamagawa number of GL(n)
Weil's conjecture, proved by Kottwitz, states that the Tamagawa number of a semisimple, simply connected algebraic group (over a number field) is 1. For example, $SL(n)$ and induced tori. Is the ...
5
votes
1answer
139 views
The $\mathbb{Q}$-rational cuspidal group of $J_0(N)$
Let $N$ be a positive integer and consider the modular curve $X_0(N)$ over $\mathbb{Q}$. Also, consider the Jacobian variety $J_0(N)$ of $X_0(N)$, which is an abelian variety defined over $\mathbb{Q}$....
2
votes
0answers
94 views
Non-cuspidal Hecke eigenforms and Eisenstein series
It's a direct check that ${\displaystyle E_{2k}(z )=\frac{\zeta(1-2k)}{2}+\sum _{n=1}^{\infty }\sigma _{2k-1}(n)q^{n}}$ is an eigenform for every Hecke operator $T_n$ with eigenvalue $\sigma_{2k-1}(n)$...
11
votes
0answers
191 views
Deligne's theorem on finite flat group schemes and generalizations
Recall Deligne's theorem that for a finite flat commutative group scheme $G$ of order $n$, the multiplication by $n$ map $[n]: G \to G$ is the zero map.
I have seen the proof a few times but I can't ...
3
votes
1answer
133 views
Classes of hyperplane sections in cohomology
Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$.
Is it true that the vector space $H^{2k}(X,\mathbf{Q}_{\ell}(k))^G$ has always positive ...
6
votes
1answer
185 views
Frobenius eigenvalues algebraic numbers
Let $X$ be a smooth projective variety over $\mathbf{F}_q$ and $\overline{X}$ its base change to $\overline{\mathbf{F}_q}$.
By Deligne’s Weil I, the eigenvalues of the geometric Frobenius acting on $...
7
votes
1answer
211 views
Schoof's Algorithm for Hyperelliptic curves over $\mathbb{F}_q$ : Question regarding computation of resultant: Gaudry
I am new to StackExchange and I am currently going through Gaudry's paper on counting points on hyperelliptic curves (see https://hal.inria.fr/inria-00512403/document). As a part of the generalization ...
4
votes
1answer
154 views
When is this localization map injective, if at all?
Let $K$ be a number field and $E$ be an elliptic curve defined over $\mathbb{Q}$. Consider the localization map
$$
E(K)\otimes \mathbb{Q}_p/ \mathbb{Z}_p \rightarrow \bigoplus_{v|p} E(K_v)\otimes \...
0
votes
0answers
102 views
Lifting singularities to zero characteristic
Suppose we have a proper (or projective) singular variety $X_0$ over a field of characteristic $p$, $k$, and with dual graph $\Gamma$. We may suppose that the components of $X_0$ are smooth and ...
4
votes
1answer
417 views
Prerequisites for reading papers of arithmetic such as Ribet, Mazur, Faltings, Wiles
I've studied some fundamentals of algebraic geometry and number theory, and now I want to read papers which seem to be the "main stream" of frontier research on arithmetic.
I've heard that Mazur's "...
1
vote
0answers
50 views
Endomorphism rings of deformations of a height $h$ formal group law
Let $k$ be an algebraically closed field of characterstic $p$, $H_0$ be a height $h$ formal group law over $k$. For any complete noetherian $W(k)$ algebra with residue field $k$, we can consider the ...
14
votes
1answer
359 views
What is the automorphic interpretation of the Weil conjectures over finite fields
I am very much a beginner in the theory of automorphic forms and I might (will?) make mistakes in what follows. Please correct me.
A loose interpretation of the Langland's philosophy is that to any ...
1
vote
0answers
42 views
Finite generation for a restricted ramification idele module
Let $k$ be a number field, let $\bar k \subseteq \mathbb{C}$ be a fixed algebraic closure of $k$, and let $S$ be the set of infinite primes of $k$. Denote by $k_S$ the maximal extension of $k$ inside $...
6
votes
1answer
220 views
How to visualize local complete intersection morphisms?
As the question title asks for, how do others visualize local complete intersection morphisms? My experiment in asking people in real life didn't pan out, so I'm consulting the MO algebraic geometry ...
10
votes
1answer
467 views
Euler factors of L-function at bad primes
This is of course a very-well known problem, but still let me ask the questions my way. Let $L(s)$ be a "motivic" $L$-function, whatever that means: in particular, it has an Euler product (including ...
6
votes
0answers
153 views
An abelian variety has good reduction $\iff$ the Neron model is proper
Let $R$ be a D.V.R. with the fraction field $K$, $A$ a $K$- abelian variety, $\mathfrak{A} \to \operatorname{Spec}R$ the Neron model of $A$.
We say $A$ has good reduction if there exists a smooth ...
12
votes
2answers
516 views
Defining abstract varieties and their morphisms over a finitely generated subfield of the base field
Let $k$ be an algebraically closed field.
By a finitely generated subfield of $k$ I mean a subfield $k_0\subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $\mathbb Q$ ...
1
vote
0answers
193 views
Hardy-Littlewood vs heuristics on the zeta zeros
The first Hardy-Littlewood Conjecture asserts:
Conjecture 1: Fix integers $0< a_1 < \ldots < a_k$. The density of those primes $p\le x$ such that $p + 2a_1,\ldots, p+2a_k$ are also primes, ...
0
votes
1answer
95 views
Smooth loci and formal neighborhoods
Let $R$ be a Noetherian local ring with maximal ideal $I$.
Suppose we have a morphism of smooth $R$-algebras $f : A\to B$ such that its reduction modulo $I^n$
$$f_n : A/I^n \to B/I^n$$
is an ...
9
votes
0answers
194 views
Zeros of $p$-adic power series and rationality
Let $K$ be a non-archimedean field with valuation ring $(V,\mathfrak{m})$, and $K\langle t_1,\ldots, t_n\rangle$ a Tate algebra of convergent power series.
Fix $f \in V\langle t_1,\ldots, t_n\rangle$....
2
votes
0answers
124 views
Solving solutions to systems of polynomial equations over $\mathbb Z$
Macaulay Resultants help identify common root in $\mathbb P^{n-1}(\mathbb K)$ of $n$ homogeneous polynomials in $n$ variables when $\mathbb K$ is algebraically closed. Is it possible that some type of ...
10
votes
2answers
625 views
How to visualize finiteness of class number?
As the question title asks for, how do others "visualize" the finiteness of class number with algebro-geometric insight? I just think of it as a result in algebraic number theory and not one in ...
4
votes
1answer
269 views
Submersion implies many rational points in image?
Let $A \colon V \to W$ be a surjective linear map
(defined over $\mathbb{Z}$),
inducing a projection
$\alpha \colon \mathbb{P}(V) \to \mathbb{P}(W)$.
Let $X \subseteq \mathbb{P}(V)$ and $Y \...
2
votes
0answers
162 views
Is there a Hodge structure for smooth proper varieties over $\mathbb{C_p}$? [duplicate]
For smooth proper varieties over $\mathbb{Q_p}$, we have several comparison theorems in p-adic Hodge theory, in particular a p-adic Hodge structure.
Now for $\mathbb{C_p}$, is there any such results ...
3
votes
0answers
116 views
Is there a way to reduce this problem to two variables through functions coming from arithmetic?
Consider following diophantine equation in $\mathbb Z[x,y,z]$ in three integer variables $x,y,z$
$$x^2+L(y,z)x+L_1(y)L_2(z)=0$$ where $L(y,z)$ is a non-homogeneous linear polynomial in $y,z$ and $L_1(...
2
votes
0answers
122 views
The growth of class number in $\mathbb{Z}_p$-extensions of function fields
Let $X$ be a curve (proper, smooth, ...) over a finite field $\mathbb F_q$ where $q$. Suppose also that $\mathbb F_q$ contains the $p$-th roots of unity, in this case we have the following (unique) ...
4
votes
1answer
228 views
Interpolation of families of local fields
Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...
11
votes
1answer
309 views
“Algebraization" of $p$-adic fields
Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.
Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion ...
3
votes
0answers
73 views
Differential of p-divisible groups
Setting: Let $p$ be a prime number and let $S$ be a scheme such that $p$ is locally nilpotent on $\mathcal O_S$ ($p^N=0$). Let $X$ be a $p$-divisible group over $S$. Let $X[p^n] $ be the kernel of ...
6
votes
0answers
191 views
Galois actions on cohomology rings of algebraic varieties
Let $k$ be an arithmetic field. Let $G_k$ be its absolute Galois group.
$G_k$ is often studied via its linear action on cohomology (etale, crystalline, ...) "groups" of algebraic varieties over $k$.
...
7
votes
1answer
237 views
analogue of Theorem of Mattuck for Abelian varieties over $\mathbf{F}_q(\!(t)\!)$
By a theorem of Mattuck [Abelian Varieties over $p$-Adic Ground Fields, Annals of Mathematics, Second Series, Vol. 62, No. 1 (Jul., 1955), pp. 92-119], for an Abelian variety $A$ of dimension $g$ over ...
8
votes
1answer
205 views
Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties
$\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\F}{\Bbb F}
\newcommand{\p}{\mathfrak{p}}
$
Let $A$ be an abelian ...
7
votes
1answer
371 views
Vector space objects in schemes - confusion
Let $R$ be the ring $\mathbf{C}\times\mathbf{C}$, and consider the affine line $\mathbf{A}^1_R$.
$\mathbf{A}^1_R$ can be given the structure of additive group scheme over $R$, denoted $(\mathbf{G}_a)...
9
votes
0answers
204 views
Examples for a conjecture of Beilinson
Beilinson has conjectured that for a regular, complete, geometrically irreducible curve $C$ of genus $g$ over a number field $k$, $rank(K_2(C))=g[k:\mathbb{Q}]$. As far as I know it is not known in ...
4
votes
0answers
137 views
Harder-Narasimhan over arbitrary coefficients
Let $X$ be an $n$ dimensional smooth projective variety over $k$. Let $H$ be a hyperplane section. Define the slope $\mu(E)=\frac{c_1(E).H^{n-1}}{rank(E)}$ for vector bundles on $X$. Does the Harder-...
9
votes
1answer
188 views
Existence of certain endomorphism of supersingular elliptic curve
Let $E$ be a supersingular elliptic curve over an algebraically closed field $K$ of characteristic $p$. Let $R = \operatorname{End}(E)$ be its ring of endomorphisms. Then, it is known $R \otimes_{\...
6
votes
0answers
267 views
Embeddings of number fields into $\mathbb{C}$ or $\bar{\mathbb{Q}}_l$
When studying arithmetic Galois representations for a number field $F$ one often fixes at the outset an embedding of its algebraic closure $\bar{F}$ into $\mathbb{C}$ or $\bar{\mathbb{Q}}_l$ and ...
