# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

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**1**answer

545 views

### Applications of integral p-adic Hodge theory

What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $\mathbb{Z}_p$-...

**4**

votes

**1**answer

244 views

### rank of Jacobian of Fermat curve and Chabauty-Coleman method

Consider the fermat curve $F(p)$ over $\mathbb Q$ which is the projective closure of $X^p+Y^p=1$ inside projective plane, where $p$ is a prime number and without loss of generality we assume $p>2$. ...

**4**

votes

**1**answer

147 views

### Morphism of sites and abelian sheaf cohomology

Let $f : \mathcal{C}\to\mathcal{D}$ be a morphism of sites (see the Stacks Project) with induced morphism of topoi
$$(f^{-1}, f_*) : Sh(\mathcal{D})\to Sh(\mathcal{C}).$$
By assumption, $f^{-1}$ is ...

**5**

votes

**0**answers

173 views

### Regular functions vs holomorphic functions

Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.
Is ...

**14**

votes

**1**answer

592 views

### GAGA for henselian schemes

In this paper, F. Kato recollects basic facts on henselian schemes and proves some partial results towards GAGA in the context of henselian schemes.
Let $I$ be a finitely generated ideal in a ...

**5**

votes

**2**answers

141 views

### Endomorphism ring of $J_0(p)$ and Hecke operators

Let $J_0(p)$ be Jacobian of the modular curve $X_0(p)$ over $\mathbb Q$ where $p$ is a prime, consider the subring $\mathbb T$ inside $End_{\mathbb Q}(J_0(p))$ generated by Hecke operators $T_n$ for ...

**4**

votes

**1**answer

472 views

### What are “arithmetic curves”?

Let $C$ be a separated irreducible reduced curve which is quasi-finite over $\mathrm{Spec}\: \mathbb Z$. Is it necessarily affine i.e. $\mathrm{Spec}\: \mathcal O$ where $\mathcal O$ is an order in a ...

**3**

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119 views

### Are there three ordinary elliptic curves $E$, $E_1$, $E_2$ such that $E^2 \cong E_1 \!\times\! E_2$?

Consider the elliptic curve $E\!: y^2 = x^3 + 1$ of $j$-invariant $0$ over an algebraically closed field $k$ of characteristics $p$. Let me remind that $E$ is ordinary (i.e., non-supersingular) iff $p ...

**1**

vote

**1**answer

184 views

### Swan-conductor and base change

Let $C$ be a proper smooth curve over a perfect field $K$ of positive characteristic $p$, $u: U \hookrightarrow C$ strictly open and $\mathfrak{F}$ a lisse (lcc) $\mathbb{F}_l $-sheaf $(l \neq p)$ on $...

**2**

votes

**1**answer

160 views

### Vector bundles on henselian schemes

Let $X$ be a smooth and projective scheme over $\mathbf{Z}_p$.
We call $\mathfrak{X}$ the ringed space whose topological space is the topological space of the special fiber of $X$, and whose ...

**4**

votes

**1**answer

208 views

### Does there exist a curve which avoids a given countable union of small subsets?

Let $X$ be a projective variety over $\mathbb{C}$. Let $X_1, X_2, \ldots$ be proper closed subsets of $X$. Then $\cup_i X_i \neq X(\mathbb{C})$. However, I am interested in a stronger statement.
...

**4**

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**0**answers

96 views

### Computation of the Newton polygon of formal modules under unramfied condition

When reading a very short paper "A generalization of the Chowla-Selberg formula" which is avaliable at http://www.math.titech.ac.jp/~taguchi/bib/cs.pdf by Yukiyoshi Nakkajima and Yuichiro Taguchi, ...

**10**

votes

**1**answer

436 views

### Deligne conjecture without Langlands correspondence

Let $X$ be a normal variety over a finite field $F_q$. Fix a prime number $l$ relatively prime to $q$. Let $\sigma$ be a irreducible lisse $l$-adic sheaf on $X$ whose determinant has finite order. It ...

**2**

votes

**0**answers

66 views

### Points of low height and low degree

Let $K$ be a number field, and let $C$ be an algebraic curve of genus $g \geq 2$ defined over $K$. We define $d_C$ to be the positive integer with the following property: there exists at most finitely ...

**14**

votes

**1**answer

383 views

### Shimura's construction of an abelian variety from cusp forms of weight $2k$

Let $\Gamma \subset \mathrm{SL}_2(\mathbb{Z})$ be an arithmetic subgroup, and $S_{2k}(\Gamma)$ the space of holomorphic cusp forms of weight $2k$ for $\Gamma$.
Let $\rho_1: \Gamma \rightarrow V_1$ ...

**4**

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140 views

### lemma II.2.4 in Harris-Taylor (about drinfeld-katz-mazur level structure on 1-dimensional $p$-divisible groups)

Lemma II.2.4 on page 82 in Harris and Taylor's "The Geometry and Cohomology of Some Simple Shimura Varieties" (or lemma 3.2 here), says that given a Drinfeld(-Katz-Mazur) level structure $\alpha:(p^{-...

**21**

votes

**1**answer

494 views

### What automorphic forms are expected to occur in the zeta function of moduli space of curves?

Assume $g \geq 1$ and $n \geq 0$, the moduli stack ${\mathcal {M}}_{g,n}$ classifies families of smooth projective curves of genus $g$ with $n$ marked points , together with their isomorphisms. It has ...

**12**

votes

**1**answer

362 views

### Some basic questions on crystalline cohomology

Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ and ${X}$ its base change to an algebraic closure $k$ of $\mathbf{F}_q$.
Crystalline cohomology $H^*_{\rm cris}(X) := H^*((X/W(k))_{\rm ...

**1**

vote

**0**answers

93 views

### finite $p$ extensions on adjoining $p$-torsion points of an elliptic curve

Let $K$ be a fixed number field and $E$ be any elliptic curve over $K$. When we adjoin to $K$ the $p$-torsion points $E[p]$, we obtain an extension whose Galois group can be embedded in $GL(2, \mathbb{...

**2**

votes

**0**answers

137 views

### Structure of the End group scheme of an abelian scheme over ring of integers

Let $O$ be the integer ring of a p-adic field $K$ (finite extension of $\mathbb Q_p$), $\mathscr{A}$ be an abelian scheme over $S=\operatorname{Spec O}$, consider the group endohomorphism scheme of $\...

**10**

votes

**1**answer

430 views

### Functoriality of crystalline cohomology

Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth projective $k$-variety.
Denote by $(X/W_n(k))_{\rm cris}$ the small crystalline site of $X$.
Let $f : X\to Y$ be a morphism of ...

**3**

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**0**answers

130 views

### Maximum number of integral roots in degree $d$ polynomial?

Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that
Each coefficient is bound in absolute value by $B$
Degree of each variable in any monomial is bound by $d$
Total degree is $d'$
$f(x_1,\...

**3**

votes

**0**answers

96 views

### When is Tate module of a semiabelian variety over a number field semisimple?

When is the $\ell$-adic Tate module of a semiabelian variety $A$ over a number field $K$ semisimple as a representation of $Gal(K^{alg}/K)$?
If $A$ is the product of a torus with an abelian variety, ...

**1**

vote

**0**answers

82 views

### Height variation of abelian varieties within an isogeny class

Let $A$ be an abelian variety defined over a number field $K$ of dimension $g \geq 2$, and put $h_F(A)$ for the (stable) Faltings height of $A$. It is well-known from the seminal paper of Faltings ...

**16**

votes

**2**answers

685 views

### revisiting $THH(\mathbb{F}_p)$

Reading through Bhatt-Morrow-Scholze's "Topological Hochschild Homology and Integral p-adic Hodge Theory" I encountered the following statement.
We use only “formal” properties of THH throughout ...

**44**

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**0**answers

2k views

### What is Prismatic cohomology?

Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...

**3**

votes

**2**answers

186 views

### Algebraic points on a curve with small degree

Let $d \geq 2$ be a positive integer, and let $K_d$ denote the compositum of all fields of degree $d$ over $\mathbb{Q}$.
Let $Y$ be an algebraic curve defined over the rationals and has genus $g \...

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**0**answers

91 views

### Variety Isomorphism Problem for Abelian Surfaces

This is a special case of this question, where it is asked whether there exists an algorithm to determine whether two varieties are isomorphic. There, an answer by Bjorn Poonen explains how to solve ...

**6**

votes

**1**answer

354 views

### Functoriality for $\ell$-adic cohomology - a question

This should a be basic enough question, but I’m a little confused.
In proving that $H^*(X,\mathbf{Q}_{\ell})$ is functorial (in the sense of Weil cohomology theories: see axiom D2 here) as $X$ ranges ...

**2**

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176 views

### Point of smallest height on an algebraic curve

Let $C$ be an algebraic curve of genus $g \geq 2$ defined over a number field $K$, and we suppose that $C(K) \ne \emptyset$ ($C$ may very well be defined over a proper subfield of $K$, but perhaps ...

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115 views

### Moduli interpretation of the integral anticanonical tower

This question is related to my reading of On torsion in the cohomology of locally symmetric varieties by Scholze.
In chapter $3$, using the theory of canonical subgroup, he produces Frobenius maps ...

**6**

votes

**1**answer

448 views

### Quaternion algebra actions on $\ell$-adic cohomology

Let $E$ be a supersingular elliptic curve over $\mathbf{F}_p$, and $H$ its endomorphism algebra $\text{End}(E)\otimes_{\mathbf{Z}}\mathbf{Q}$, a quaternion algebra (non split at $p$ and $\infty$).
...

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109 views

### Universal elliptic curve over anticanonical tower

While I'm reading Scholze's paper "On torsion in the cohomology of locally symmetric varieties" he constructs the anticanonical tower passing through the construction of an integral model $X_{\infty}$ ...

**4**

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178 views

### Carayol's “ramified Eichler-Shimura relation” and its applications

In his paper "Sur la mauvaise reduction des courbes de Shimura" from '86 H. Carayol shows the following congruence relation:
Let $M$ be the tower of Shimura curves over a totally real $F$, associated ...

**3**

votes

**1**answer

96 views

### Comparing the height of a curve and a singly branched cover

Let $C$ be an algebraic curve of genus $g \geq 2$ defined over a number field $K$, having good reduction outside of a finite set $S$ of primes in $K$. A singly branched cover $C'$ of $C$ is a curve ...

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79 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**

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184 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**

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**0**answers

86 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**

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302 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 $A/\mathfrak{m}^n$ ...

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163 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

**1**answer

187 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**

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131 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 $2g$, ...

**18**

votes

**1**answer

371 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

**0**answers

91 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

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343 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

**1**answer

143 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)?

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243 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

**1**answer

352 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

**0**answers

157 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

**1**answer

171 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 $\mathbb{Q}$....