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Questions tagged [arithmetic-geometry]

A subject that lies at the intersection of algebraic geometry and number theory dealing with varieties, the Mordell conjecture, Arakelov theory, and elliptic curves.

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What condition on an abelian variety ensures that the associated formal group has integral coefficients?

I am new in the study of abelian variety or in general algebraic variety. I am looking for a sufficient condition that ensures the formal group associated with an abelian variety has integral ...
NT2024's user avatar
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Twist of elliptic curve by degree $n$ extension

Let $E/K : y^2=x^3+ax+b$ be an elliptic curve over number field $K$. For quadratic extension $L=K(\sqrt{D})/K$, $E_D/K : Dy^2=x^3+aX+b$ is called a twist of $E/K$ by $D$. This curve $E_D$ has ...
Poitou-Tate's user avatar
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Computing degree of $x$ map for elliptic curve given by Weierstrass equation

Suppose $E$ is an elliptic curve given by the Weierstrass equation $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 $$ I want to calculate the degree of the map $$ \varphi\colon E\to\mathbb{P}^1\qquad\quad[x,y,1]...
Navid's user avatar
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3 votes
1 answer
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Proof of theorem $V.4.1(c)$ in Silverman Elliptic Curves, How to Deduce $H_p(t)$ has no multiple roots?

I am trying to understand the proof of theorem $V.4.1(c)$ in Silverman Arithmetic of Elliptic Curves. Let $p$ be a prime, $q=p^n$, and $m=(p-1)/2$. Define the polynomial $$H_p(t)=\sum_{i=0}^m \binom{m}...
Snacc's user avatar
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1 answer
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Isomorphism as $\Bbb{F}_p$ vector space and Proof of local Tate--Duality $H^1(G_K,E)[p]\cong (E(K)/pE(K))^*$

This is a question regarding Theorem $1.4$ of https://kskedlaya.org/kolyvagin-seminar/duality.pdf. Let $E/K$ be an elliptic curve over number field $K$. Let $p$ be a prime number. The goal of Theorem $...
Poitou-Tate's user avatar
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1 answer
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Relation between direct sum $\bigoplus$ and restricted product $\prod'$ of Galois cohomology

This is a question about the relation between directed sum $\bigoplus_{v \in M_K} H^1(\text{Gal}(\overline{K}/K), M)$ and restricted direct product $\prod'_{v \in M_K} H^1(\text{Gal}(\overline{K}/K), ...
Poitou-Tate's user avatar
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64 views

Accessible result motivating Arithmetic Geometry

My understanding of arithmetic geometry is that it applies concepts originating from algebraic geometry to schemes that are quite different from these initially studied in this field (those associated ...
Weier's user avatar
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$\mathbb{F}_1$-schemes under $\text{Spec}_\mathbb{Z}$

$\text{Spec}_\mathbb{Z}$ is the terminal object in the category of schemes. In the context of Borger's absolute geometry, we can treat the category of schemes as a subcategory of "the category of ...
L. E.'s user avatar
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Silverman AEC Exercise 7.4

The following problem in chapter 7 of Silverman's book has been bothering me: Here's my progress: I first tried to solve it in the $\text{char}(k) \ne 2,3$ case, wherein we can assume the equation is ...
Aditya Khurmi's user avatar
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1 answer
106 views

Is $H^1(G_{\Bbb{Q}_p},E[2])=0$ for good prime of $E/\Bbb{Q}$?

Let $E/\Bbb{Q}$ be an elliptic curve over number field $\Bbb{Q}$. Let $p$ be a prime number. $H^1(G_{\Bbb{Q}_p},E)$ is $0$ for good prime of $E/\Bbb{Q}$ because every genus $1$ curve over $\Bbb{F}_p$ ...
Poitou-Tate's user avatar
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2 votes
1 answer
126 views

Does $\#\text{Ker}(f^*)\le \#\text{Im}(f)$ hold? Duality of profinite group

Let $M, M'$ be a profinite groups. Let $M^*=\text{Hom}_{conti}(M,\Bbb{Q}/\Bbb{Z})$ be a dual of $M$. Let $f:M\to M'$ be a homomorphism of abelian group. Let $f^*: M'^*\to M^*$ be a map defined by $g\...
Poitou-Tate's user avatar
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1 answer
65 views

What is the order of $\text{Ker}(\hat{f})$ when order of annihilator of $\text{Ker}f$ is given?

Let $A,B$ be an abelian group. $\hat{A}=\text{Hom}(A,\Bbb{Q}/\Bbb{Z})$,. Let $f$ be a map $ f : A\to B$, and $\hat{f}: \hat{B}\to \hat{A}$ be dual of $f$. In general, $\mathrm{Im}(f)$ is the ...
Poitou-Tate's user avatar
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2 votes
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Scheme theoretic reduction map for elliptic curves

Consider $K$ a local field with $\text{char }k=0$ and $A$ an abelian surface over $O_K$. Let $A[2]$ be the 2-torsion finite flat group scheme over $O_K$ and $A[2]^\circ$ be the connected component of ...
Ja_1941's user avatar
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'Galois cohomology of elliptic curves' , Poitou-Tate sequence

This is a question about exact sequence of p5 of 'Galois cohomology of elliptic curves' by Coates Sujatha which is called Pitou-Tate exact sequence. Let $F$ be a number field. Let $S$ be a finite set ...
Poitou-Tate's user avatar
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1 vote
0 answers
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In a finite field extension $L/K$, is it true that the order of Selmer group $\text{Sel}^2(E/K)$ does not decrease?

Let $E/K$ be a fixed elliptic curve over nuber field $K$. Let $\text{Sel}^2(E/K)$ be a Selmer group of $E/K$. In a finite field extension $L/K$, is it true that the order of $\text{Sel}^2(E/K)$ does ...
Poitou-Tate's user avatar
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1 vote
1 answer
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The arithmetic-geometric mean of Gauss

The arithmetic-geometric mean of Gauss. Let $0 < a < b$. We define the two sequences $a_n$ and $b_n$ as follows: $a_0 = a, b_0 = b, a_{n+1} = (a_n b_n)^{1/2}, b_{n+1} = \frac{a_n + b_n}{2}$ ...
xoxo's user avatar
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Cohomology of number fields, Theorem (8.7.9), $Ш^1(G_S,A)\to H^1(G_S,A)\to \prod_{p\in S} H^1(k_p,A)\to \hat{H^1(G_S,A')}$

I'm reading a book 'Cohomology of number fields' Second edition by J.Neukirhi A.Schmidt, K.Winberg. In the page 511, theorem (8.7.9), there appears suddenly. $Ш^1(G_S,A)\to H^1(G_S,A)\to \prod_{p\in S}...
Poitou-Tate's user avatar
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2 votes
1 answer
87 views

Group action on cohomology is trivial

Let $G$ be a group and $H$ be a normal subgroup. Let $M$ be $H-$module. I want to check $G/H$ acts on $H^1(H,M)$ by $(\bar{\sigma}*X)(g):=\sigma X({\sigma}^{-1}g\sigma)$ where $X $ is cocycle in $H^1(...
Poitou-Tate's user avatar
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3 votes
0 answers
56 views

Proof of $\text{Cor} \circ \text{Res}=[n]$ for group cohomology

Let $G$ be an abelian group and $M$ a $G$-module. The basic definitions: Let $H < G$ be a subgroup of finite index $n$. We have a map $tr: H^0(H, M) \rightarrow H^0(G, M)$ on group cohomology ...
Poitou-Tate's user avatar
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0 votes
0 answers
61 views

The definition of action of $G/H$ on group cohomology $H^1(H,M)$

Let $G$ be a group and $H$ be a normal subgroup of $G$. Let $M$ be $G$-module. Let $H^1(H,M)$ be first group cohomology. $G$ acts on $H^1(H,M)$ by $(\sigma*f)(g):=gf({\sigma}^{-1}g{\sigma})$・・・①. My ...
Poitou-Tate's user avatar
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6 votes
2 answers
218 views

Kernel of restriction and cokernel of corestriction of group cohomology

Let $G$ be an abelian group and $M$ a $G$-module. The basic definitions: Let $H < G$ be a subgroup of finite index. We have a map $tr: H^0(H, M) \rightarrow H^0(G, M)$ on group cohomology defined ...
Poitou-Tate's user avatar
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0 votes
0 answers
92 views

Philosophy of applying Faltings' product theorem

Faltings' product theorem says that on $\mathbb{P}=\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_m}$ over $k$ of characteristic $0$. For $\epsilon>0$, $d_1> \cdots >d_m$ decrease ...
finiteness's user avatar
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0 answers
38 views

Semistable reduction of Elliptic curves

Let $E$ be an Elliptic curve defined over $\mathbb{Q}$ and $p$ be a prime at which $E$ has semistable reduction i.e. either good reduction or multiplicative reduction. Let's see the same $E$ as an ...
Kalas678's user avatar
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0 votes
1 answer
32 views

Height and intersection

Let $X$ be a surface. Let $C \subset X$ be a curve on $X$ and $P$ an algebraic point on that curve. Take an effective divisor $D$ on $X$, one can define the height of $P$ w.r.t. $D$, $h_D(P)$. Now if $...
finiteness's user avatar
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0 answers
45 views

Some questions about almost étale extension of local field

I'm in trouble understandig the example and proof of Example 4.2.2 in Denis Benois's lecture note about $p$-Adic Hodge Theory, he gives the following definition: A finite extension $E/F$ of non-...
Kevin's user avatar
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1 vote
1 answer
56 views

References for Weil restriction?

I'm reading some people's notes and some fantastic answers on stackexchange on Weil restriction. I don't have a background in algebraic geometry, so some of the simplified versions, particularly of $\...
Batrachotoxin's user avatar
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0 answers
33 views

Is the Weil pairing on elliptic curves a perfect pairing

Let $E/K$ be an elliptic curve and $m$ be a positive integer such that char$(K)\nmid m$. I know that the Weil pairing defined on elliptic curves is a bilinear & non-degenerate pairing. However is ...
user631874's user avatar
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0 answers
25 views

horizontal divisor and normalization

Let $X$ be a regular arithmetic surface over a number field $k$, let $P \in X(\bar{k})$ be an algebraic point. Let $K=k(P)$, the defining field of the point $P$. Let $B'$ be the arithmetic curve ...
finiteness's user avatar
-1 votes
1 answer
86 views

Does $\text{Aut}(E)\cong E$ hołd for elliptic curves?

Let $E/K$ be an elliptic curve over a field $K$. Let $Aut(E)$ be a group of all automorphism of $E$ as algebraic curves. Then, does $AutE\cong E$ holds ? There is a natural injection $E\to \text{Aut}(...
Poitou-Tate's user avatar
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1 vote
0 answers
73 views

Using étale fundamental group to show unramifiedness of Tate module

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with good reduction at $p$, i.e., there exists an elliptic curve $\mathcal{E}$ over $\mathbb{Z}_p$ whose generic fiber is isomorphic to $E$. I have ...
WLOG's user avatar
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0 votes
0 answers
47 views

Pair of torsors and divisors and Galois cohomology

Let $E/\Bbb{Q}$ be an elliptic curve. $H^1(G_{\Bbb{Q}},E[2])$ is bijection with the set of pair of torsors and divisor of degree $2$. Let call the latter set $WCD(E/\Bbb{Q})$. I tried to construct a ...
Poitou-Tate's user avatar
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1 vote
0 answers
61 views

$\text{Im}(f)=\text{ker}(\widehat{f})$, written in 'Galois cohomology of elliptic curves' by Coates and Sujata

Let $K$ be a number field and $K_v$ be the completion of $K$ at the place $v$. Consider an elliptic curve $E/K$ over $K$. Let $f: H^1(G_K,E[2])\to \bigoplus_v H^1(G_{K_v},E[2])$ be a natural map. For $...
Poitou-Tate's user avatar
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1 vote
0 answers
38 views

Application of the seesaw principle

The seesaw principle says in general the following. If $X,T$ are varieties with $X$ complete, and $\mathcal{L}$ a line bundle on $X\times T$, then $$ T_{1} = \{t\in T: \mathcal{L}_{X\times\{t\}} \...
user758193's user avatar
1 vote
1 answer
47 views

Isomorphism $(A\times B)^\vee\to A^\vee\times B^\vee$ for abelian varieties

Let $A$ and $B$ be abelian varieties, and consider the natural map $$f:(A\times B)^\vee\to A^\vee\times B^\vee$$ sending a line bundle on $A\times B$ to the restrictions to $A\times\{0\}$ and $\{0\}\...
user246336's user avatar
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0 votes
0 answers
24 views

Mordell-Weil theorem for rational points on $y^2=x^2+1$

The set $R$ of rational points on the curve $y^2=x^2+1$ can be viewed as a group under the following operation: if $X_1=(x_1,y_1),\ X_2=(x_2,y_2)\in R$, define $X_1*X_2:=(x_1y_2+x_2y_1,x_1x_2+y_1y_2)$....
aleph0's user avatar
  • 103
2 votes
0 answers
115 views

Notion of functions on Elliptic curves over scheme and writing them as power series.

I am trying to understand a statement in Andrew Snowden's Notes regarding the representability of the functor $F_{\Gamma(3)}$ (Section 14.1). Here, $F_{\Gamma(N)}(S)= \{isom \; classes \; of \; (E,(P,...
Math_user's user avatar
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0 votes
0 answers
27 views

Local Tate pairing on elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and $k_n=\mathbb{Q}_p({\zeta_{p^n}})$. Let $T$ be the $p$-adic Tate module of $E$. Then there is a local Tate pairing, $$<,>_n: H^1(k_n, E[p^\...
user631874's user avatar
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0 answers
36 views

rational elliptic curve: torsion subgroup defined over abelian extensions

I'm thinking about a question, that is if E is a rational elliptic curve, K is a cyclotomic field, how can I find the minimal subfield of K such that the torsion subgroup over K can also be defined ...
Handle135790's user avatar
0 votes
1 answer
68 views

Profinite completion of local Mordell-Weil group

Let $K$ be a finite extension of $\Bbb{Q}_p$. Let $E/K$ be an elliptic curve over $K$. Let $E(K)$ be the Mordell-Weil group of $E/K$. Let $\widehat{E(K)}$ be the profinite completion of $E(K)$, that ...
Poitou-Tate's user avatar
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0 votes
1 answer
131 views

Ramification of mod $\ell$ representation of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p,\ell$ be two prime numbers. Consider the mod $\ell$ representation $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$...
ZZP's user avatar
  • 150
1 vote
0 answers
38 views

Fix points of Galois action on etale homology

Let $k = \mathbb{F}_q$ be a finite field of characteristic $p$ and $C$ a projective, smooth curve over $k$. Denote by $\bar C$ the base change of $C$ to a separable closure $\bar k$ of $k$. Let $\ell$ ...
Erich's user avatar
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0 votes
0 answers
103 views

Elliptic curve theory : Injection from $\widehat{E(K)}$ to $E(K_v)$

Let $E/K$ be an elliptic curve over a number field $K$. Let $\widehat{E(K)}$ be profinite completion of $E(K)$.// Let $v$ be a place of $K$. Let $K_v$ be completion of $K$ at $v$. Is there an ...
Poitou-Tate's user avatar
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1 vote
0 answers
41 views

A nice description for $t^{k-1}B_{cr}^+/t^kB_{cr}^+$?

When $t$ is a uniformizer of the integral de Rham period ring, $B_{dR}^+$, there is an isomorphism $t^{k-1}B_{dR}^+/t^{k}B_{dR}^+\cong \mathbb{C}_p(k-1)$. Is there a nice description for $t^{k-1}B_{cr}...
kindasorta's user avatar
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0 votes
1 answer
191 views

Example of an elliptic curve which is not defined over $\Bbb{Q}$

An elliptic curve $E$ is defined over $\Bbb{Q}$ if only if it is isomorphic over $\overline{\Bbb{Q}}$ to an elliptic curve whose coefficients are in $\Bbb{Q}$. My question is, what are examples of an ...
Poitou-Tate's user avatar
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1 vote
1 answer
70 views

Why is the Function Field of the Modular Curve $X(N)$ defined over $\mathbb{Q}(\mu_n)$?

Following Section 7.6 in Diamond & Shurman, the algebraic model of $X(N)$ is constructed (a priori) over $\mathbb{Q}$ by first defining its function field as \begin{equation*} \mathbb{Q}(j,f_{(0,1)...
Josu P. Z.'s user avatar
1 vote
1 answer
86 views

Using Hurwitz's formula, about $\mathbb{F}_p$-rational points

Question. For $f(x) \in \mathbb{F}_p[x]$ a monic separable polynomial, define $U=\mathrm{Spec} \mathbb{F}_p[x,y]/(y^2-f(x))$. Consider the degree 2 morphism $f:U\to \mathbb{A}^1$ given by $(x,y)\...
WLOG's user avatar
  • 1,336
1 vote
1 answer
96 views

$ \mathbb{Q}_p $-admissible p-adic representaion

In Serin Hong's notes on $p$-adic hodge theory, he claimed that every $p$-adic representation is $\mathbb{Q}_p$-admissible, as $D_{\mathbb{Q}_p}$ is the identity functor.But by the definition of the $...
Kevin's user avatar
  • 395
3 votes
1 answer
106 views

Regular schemes (related to quadratic rings)

Question. Let $X=\mathrm{Spec}\mathbb{Z}[x]/(x^2-p)$. Show that $X$ is a regular scheme (i.e., all the local rings $\mathcal{O}_{X,x}$ are regular local rings) if and only if $p=2$, or $p\equiv 3\pmod{...
WLOG's user avatar
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0 votes
1 answer
74 views

Addition of $n$ points on elliptic curves over a finite field

Let $E$ an elliptic curve over a finite field $\mathbb{F}_q$. Let $P_1, \dots, P_n \in E(\mathbb{F}_q) \backslash \{\mathcal{O}\} $ be $n$ distinct rational points of $E$. Suppose there exists a ...
RiverOfTears's user avatar
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0 answers
25 views

Subvariety hitting every residue disk

Let $S$ be an irreducible projective surface defined over $\mathbb{Z}_p$, denote its special fibre by $S'$, and let $\gamma: S\longrightarrow S$ be an automorphism which is compatible with the ...
kindasorta's user avatar
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