Questions tagged [arithmetic-geometry]

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

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Meaning of dagger cohomology $H^{1 \dagger}(G^\dagger)$ in "Frobenius and Monodromy Operators" by Coleman and Iovita

Let $G$ be an abelian variety with good reduction over a finite extension $K$ of $\mathbb{Q}_p$. In equation (2.4) on page 179 of my edition of "The Frobenius and monodromy operators for curves ...
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Algebraic properties of Witt vectors $W(K^{\flat\circ})$, $K$ a characteristic 0 perfectoid field

Let $K$ be as in the title with tilt $K^\flat$. $W = W(K^{\circ\flat})$ satisfies a universal property: it is the unique $p$-adically complete $p$-torsion free $\mathbb{Z}_p$-algebra $A$ with $A / pA \...
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Is the weight-monodromy conjecture known for unramified representations?

Let $X$ be a smooth proper variety over a number field $K$, $v$ a place of $K$ lying over a prime number $p \neq \ell$, and $V := H^n(X_{\overline{K}};\mathbb{Q}_{\ell})$. Suppose $V$ is unramified at ...
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Cubic twist of elliptic curves and its rank

Let $E/\mathbb{Q}$ be an elliptic curve defined by $E: y^2 = x^3 + b$ (where $b \in \mathbb{Q}$). Let $E_D$ be an elliptic curve defined by $E_D: y^2 = x^3 + bD^2$. $E$ and $E_D$ are isomorphic over $\...
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Deformations over $A_{\inf}$

Setup: Let $K$ be a perfectoid field of characteristic $0$ with tilt $K^{\flat}$. Let $A_{\inf}=W(\mathcal{O}_{K^{\flat}})$ be the infinitesimal period ring. Let $\mathcal{X}$ be a flat, projective $\...
Kostas Kartas's user avatar
5 votes
1 answer
381 views

Cohomology of Shimura varieties before and after completion at some prime

Let $(G,X)$ be a Shimura datum with reflex field $E\subset \mathbb C$. For any neat open compact subgroup $K \subset G(\mathbb A_f)$, let $\mathrm{Sh}_K$ denote the associated Shimura variety. It is a ...
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Definition of intersection pairing on an arithmetic surface

$\def\div{\operatorname{div}} \def\Spec{\operatorname{Spec}}$Let $K$ be a number field, $O_K$ be the ring of integers, and $X \to \Spec(O_K)$ be a regular arithmetic surface. I want to understand how ...
dummy's user avatar
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2 votes
1 answer
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An example of a geometrically simply connected variety with infinite Brauer group (modulo constants)

$\DeclareMathOperator\Br{Br}$Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\Br(X)/{\Br(k)}$ being an infinite ...
Victor de Vries's user avatar
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p-adic L functions from Selmer groups - how canonical are they?

For this question, I am going to be very concrete but very much appreciate broader viewpoints. Let $F$ be a number field and define $F_n = F(\mu_{p^n})$ and let us suppose for simplicity that $\mu_p \...
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Global duality theorem for 2-part

$\DeclareMathOperator\coker{coker}\DeclareMathOperator\Sha{Sha}$Let $K$ be a number field. Let $E/K$ be an elliptic curve over $K$. Suppose finiteness of $\Sha(E/K)$. According to Global duality ...
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Conductor of determinant of a 2-dimensional Galois representation divides conductor of representation

I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I'm stuck trying to prove that $N(\det\rho)$ divides $N(\rho)...
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Maximal p-extension and pro-p extension

I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help. Q_1: About terminology $p$-extension. I find many reference use maximal $p$-extension or maximal abelian p-extension ...
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Is there an isotrivial elliptic surface of positive rank having a section of order $3$?

Let $k$ be a field of characteristic $p > 3$. I cannot find any example of ordinary isotrivial elliptic $k$-surface $E$ (i.e., elliptic $k(t)$-curve, where $t$ is a variable) whose Mordell-Weil ...
Dimitri Koshelev's user avatar
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Looking for an example of a point $P$ on an abelian variety $X$ such that no curve on $X$ contains all multiples of $P$

Is there an example of an abelian variety $X$ defined over a number field $K$, with $\dim X > 1$, and a $K$-rational point $P$ on $X$, such that no curve $C$ on $X$ (say defined over a number field)...
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Large 2-part of Tate–Shafarevich group over $\Bbb{Q}$ with small number of prime factor of discriminants

$\newcommand{\Sha}{\operatorname{Sha}}$Let $E/\mathbb{Q}$ be an elliptic curve, and let $\Sha(E/\mathbb{Q})$ denote the Tate–Shafarevich group of $E/\mathbb{Q}$. It is known that the 2-primary ...
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Multiplicity and the perfect projective line

Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$. Let $\Gamma$ be the ...
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1 answer
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Understanding an example of abelian-type Shimura varieties

I'd like some help understanding the idea of abelian-type Shimura varieties. In paricular, I understand an abelian-type Shimura datum $(G,X)$ generally parameterizes non-rational Hodge structures ...
xir's user avatar
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Self-intersection of the diagonal on a surface

Let $X$ be a smooth projective curve over the complex numbers, and take $\Delta$ the diagonal divisor on $X\times X$. Using the adjunction formula, one computes $\Delta\cdot\Delta =2-2g$ for $g$ the ...
Tim's user avatar
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168 views

Equations for conic del Pezzo surfaces of degree one

Let $X$ be a del Pezzo surface of degree one over a field $k$ of characteristic not $2$ equipped with a conic bundle $\pi: X \rightarrow \mathbb{P}^1$. By Theorem 5.6 of this paper, $X$ admits a ...
Sam Streeter's user avatar
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Global class field theory and closure of unit groups

I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
Tim's user avatar
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1 answer
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Finitely generated $\mathbb{Z}$-algebra embeds into unramified $p$-adic ring

Let $R$ be a finitely generated ring, that is, a $\mathbb{Z}$-algebra of finite type. Assume that $\operatorname{char}(R) = 0$. It follows from Noether's normalization lemma that $R$ can be embedded ...
HASouza's user avatar
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Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D/\Bbb{Q}$ is bounded by some constant $M$, for all integers $D \in \Bbb{Z}$?

Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Let $E_D$ be a quadratic twist of $E$ also defined over $\Bbb{Q}$. Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D/...
Duality's user avatar
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Extend line bundle on regular curve to it's regular model

Let $S$ be the spectrum of an excellent discrete valuation ring with field of fractions $K$ and $C$ be a proper integral regular curve over $K$. Assume, $C$ admits a proper regular flat model $\...
user267839's user avatar
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1 answer
177 views

Dimension of Zariski closure of a closed point of generic fiber

Let $S= \operatorname{Spec} A$ be a local Dedekind scheme of dimension $1$, (eg spectrum of localization at a prime of the ring of integers of a number field). Let $s \in S$ it's unique closed point ...
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Adic generic fiber of a small formal scheme in the sense of Faltings

$\DeclareMathOperator{\Spf}{Spf}\DeclareMathOperator{\Spa}{Spa}$In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic ...
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2 votes
0 answers
127 views

Isom-functor for generalized elliptic curves is representable

I am studying Deligne-Rapoport's 'Les Schémas de Modules de Courbes Elliptiques'. The following excerpt is from the proof of Theorem 2.5, Chapter III, page DeRa-61, (page DeRa-61) (*) For $C_i$, ...
ayan's user avatar
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306 views

Square-zero extensions mod $p^n$

$\DeclareMathOperator\LMod{LMod}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Sp{Sp}$A square-zero extensions of rings is, conceptually, a map of rings $R \to A$ such that any two elements in the ...
Mori B.'s user avatar
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1 answer
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Interpreting group-theoretic sentences as statements about algebraic groups

Suppose we are given a sentence in the language of groups, e.g. $\phi=\forall x\forall y(x\cdot y=y\cdot x)$, and suppose that we are also given the data defining an algebraic group $G/k$. One can ...
Nathan Lowry's user avatar
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92 views

Why is norm map on ker of reduction of elliptic curve surjective?

Let $E$ be an elliptic curve over local field $K$ of characteristic $0$. Let $F/K$ be quadratic extension. Let $E_1$ be kennel of reduction. Let $N: E(F)\to E(K)$ by $P\to P+P^{\sigma}$($\sigma$ is a ...
Duality's user avatar
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6 votes
0 answers
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Failure of injectiveness of maps between cotangent spaces of abelian varieties

Let $p$ be a prime and $K$ a finite extension of $\mathbb Q_p$ with ramification index $e$. Let $\mathcal O_K$ be the ring of integers of $K$ and $k$ its residue field and the unique maximal ideal. ...
Maarten Derickx's user avatar
2 votes
0 answers
230 views

Is there any relation between Berkovich spaces over $\Bbb Z$ and Arakelov theory?

As I understand it, both Arakelov geometry and Berkovich geometry over $\Bbb Z$ (or $\mathcal O_K$) consider geometric objects that contain in some sense information about both archimdean and ...
Lukas Heger's user avatar
1 vote
1 answer
212 views

On the estimate for a double exponential sum

I encounter a hyper-Kloosterman sum which needs some help from the experts here: For any integers $q,s \in \mathbb{N}^+$(which may not be necessarily co-prime with each other), how to bound the sum: $$...
hofnumber's user avatar
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1 answer
306 views

Pontryagin dual of cokernel, $(\operatorname{coker} F)^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $

Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve. Consider the natural map $$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{...
Duality's user avatar
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17 votes
3 answers
2k views

Are some congruence subgroups better than others?

When I first started studying modular forms, I was told that we can consider any congruence subgroup $\Gamma\subset\operatorname{SL}_2(\mathbb{Z})$ as a level, but very soon the book/lecturer begins ...
Coherent Sheaf's user avatar
1 vote
1 answer
113 views

About the power of numbers primes distribution

Let $r>0$, $p\neq q$ two primes numbers and $A=\{(m,n)\in\mathbb N^2; |p^m-q^n|\leq r\}$. Is it true that $A$ is a finite set?
Dattier's user avatar
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2 votes
0 answers
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A direct proof that every projectivity between parallel lines is affine

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
Taras Banakh's user avatar
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1 vote
0 answers
72 views

A general theory of pairings

Bilinear forms and bilinear maps for vector spaces over a field are standard material for an introductory course in linear algebra. There are also text books for bilinear forms and related quadratic ...
Thomas Preu's user avatar
1 vote
0 answers
155 views

Motivic complex on arithmetic schemes

If we believe the finite generation of motivic cohomology for regular arithmetic schemes like $X$ then we can see that (using Quillen-Lichtenbaum)for infinitely many primes $l$ we have an isomorphism ...
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11 votes
1 answer
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Does every finite affine plane have the doubling property?

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
Taras Banakh's user avatar
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1 vote
0 answers
104 views

Large Tate-Shafarevich group of an elliptic curve with the form $E_{p,n}:y^2=x^3+p^nx$

Let $p$ be a prime number and $n$ be positive integer. Let $E_{p,n}:y^2=x^3+p^nx$ be an elliptic curve. LMFDB reads in the case $(p,n)=(73,3)$ , $\#Sha(E_{p,n})=64$. This is the biggest size of $Sha(...
Duality's user avatar
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0 votes
1 answer
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Almost Pell type equation

Consider the following Diophantine equation $$ 2x^2-Ny^2 = -1. $$ where $N$ is an integer. Is there any result expressing the values of $N$ for which the above equation admits an integral solution?
Puzzled's user avatar
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Another generalisation of euclidean division on integers

Let $n \in\mathbb N^*$. What are all the surjective functions $f: \mathbb N \rightarrow \{0,...,n-1\}=E $ such that there exist functions $g,h$ from $E^2$ to $E$ with: $\forall (m,k) \in\mathbb N^2,f(...
Dattier's user avatar
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3 votes
0 answers
184 views

Tate isogeny theorem over varieties?

Let $X$ be a nice scheme, $\pi:E\to X$ an elliptic curve, and $\ell$ a prime invertible on $K$. Then we can consider the "Tate module" $(R^1\pi_*\mathbb{Z}_{\ell})^\vee=\hbox{''}\varprojlim\...
Curious's user avatar
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3 votes
0 answers
106 views

How does the number of connected components of the Néron model change in a family of abelian varieties?

Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other ...
Nathan Lowry's user avatar
2 votes
0 answers
244 views

The group of the modular automorphisms of the Shimura curves

Let $B$ be a rational indefinite division quaternion algebra, $(X,G)$ the Shimura datum associated with $B$ (i.e., $X$ is the upper half plane and $G(R) = (B \otimes_\mathbb{Q} R)^*$ for a ring $R/\...
k.j.'s user avatar
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10 votes
0 answers
780 views

How to think of algebraic geometry in characteristic p?

How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
JustLikeNumberTheory's user avatar
8 votes
2 answers
326 views

$n$-torsion fields of an elliptic curve defined over $\mathbb{Q}$

Let $E/\mathbb{Q} = E_{a,b}$ $$\displaystyle y^2 = x^3 + ax + b, a,b \in \mathbb{Z}$$ be an elliptic curve defined over the field of rational numbers, and let $n \geq 3$ be an integer. Let $K_n$ be ...
Stanley Yao Xiao's user avatar
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0 answers
55 views

Discreteness of Szpiro ratios among twist minimal curves?

For an elliptic curve $E/\mathbb{Q}$, let $D(E) = |\Delta_{\min}(E)|$ be the minimal (absolute) discriminant of $E$, that is, the smallest possible value of the discriminant running over all integral ...
Stanley Yao Xiao's user avatar
5 votes
1 answer
338 views

Fermat cubic hypersurfaces over finite fields

Consider the Fermat cubic $$ X = \{x_0^3+\dots +x_n^3 = 0\}\subset\mathbb{P}^n_{\mathbb{F}_{q}} $$ over a finite field $\mathbb{F}_{q}$ with $q$ elements. If $q \equiv 2 \mod 3$ then the projection $\...
Puzzled's user avatar
  • 8,842
4 votes
1 answer
204 views

Points on affine hypersurface over finite field

I am interested in the hypersurface $X\subset\mathbb{A}^4_{\mathbb{F}_{5^n}}$ defined by $$ X = \{x^3 + 3xy^2 + z^3 + 3zw^2 + 1 = 0\} $$ over a finite field $\mathbb{F}_{5^n}$ with $5^n$ elements. Via ...
Puzzled's user avatar
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