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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Resources for Abelian surfaces
I have recently gone through Silverman's Arithmetic of Elliptic Curves and was looking for sources on Abelian surfaces but am struggling to find anything. I wasn't expecting to find the Arithmetic of ...
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Is this sequence arithmetico geometric? [closed]
I am im desperate need of help to solve a mathematic exercise.
I have to determine if this sequence is arithmetico geometric but I struggle explaining it.. I am supposed to present it in front of my ...
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Looking for a counter-example
I read on this page that an endomorphism of degree one of a smooth algebraic variety must be an automorphism. The proof uses Zariski's main theorem.
My question is this: are there examples of ...
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Showing that the kernel of reduction map is isomorphic to group associated to formal group
This proposition is from Silverman's Arithmetic of Elliptic Curves.The proof is a bit long to type so I've included the image.
Here $K$ is a local field with residue field $k$ of its ring of integers ...
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Dual abelian variety in the case of elliptic curves
For abelian variety $A$, dual abelian variety $A'$ is defined, for example, in https://en.wikipedia.org/wiki/Dual_abelian_variety.
But what is the case $A=E$ is an elliptic curve?
In particular, does $...
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Statement of Golfeld's conjecture
Goldfeld conjecture asserts that
for elliptic curve $E/\Bbb{Q}$,
$rank(E_D/\Bbb{Q})=0$ for $50%$ square free D's
$rank(E_D/\Bbb{Q})=1$ for $50%$ square free D's
$rank(E_D/\Bbb{Q})=2$ for $0％$ square ...
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Is there an elliptic curve with a bounded rank for all of its quadratic twists?
Is there an elliptic curve with a bounded rank for all of its quadratic twists?
Consider the elliptic curve $E/\mathbb{Q}$ defined by:
$y^2 = x^3 + ax + b$.
Its quadratic twist, $E_D$, is given by $...
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The equivariant BSD conjecture (and the $\rho$-isotypical component)
I am trying to understand the statement of the equivariant BSD conjecture.
Let $E/\mathbb{Q}$ be an elliptic curve. Let $\rho$ be a finite-dimensional irreducible Artin representation, and let $K/\...
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Is it difficult to determine the upper bound of the rank of the quadratic twist of a given elliptic curve?
Let $E:y^2=x^3+ax+b$ be a given elliptic curve over $\mathbb{Q}$, and let $D$ be an integer. The quadratic twist of $E$, denoted by $E_D$, is given by the equation $E_D:Dy^2=x^3+ax+b$. A natural ...
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Difference between $E(\overline{K})$ and $E$ for elliptic curve $E/K$
Let $K$ be a field, and consider an elliptic curve $E/K$. Let $\overline{K}$ be algebraic closure of $K$.
It is common to abbreviate or identify $E(\overline{K})$ with $E$.
Can this abbreviation or ...
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Quasi periodic entire function [duplicate]
This problem is exercise 6.1 from Silverman's Arithmetic of Elliptic Curves.
6.1 Let $\Lambda=\mathbb{Z}\omega_1$+$\mathbb{Z}\omega_2$ be a lattice.Suppose $\theta(z)$ is an entire function,with the ...
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Why $Sha(E_n/\mathbb{Q})[2] = 0 \implies Sha(E_n/\mathbb{Q})[2^\infty] = 0$ holds?
Let $A$ be an Abelian group. Define $A[2^n] = \{a \in A \mid 2^na = 0\}$ for a positive integer $n$.
Then, let $A[2^\infty] = \bigcup_{n\geq 1} A[2^n]$.
In general, $A[2]=0 \nRightarrow A[2^\infty]=0$....
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Is rank of $E_p:y^2=x^3+px$($p$ is a prime number) still open?
In 'Rational Points on Elliptic Curves' by Silverman and Tate, it is stated that the rank of the elliptic curve $E_p: y^2 = x^3 + px (p \equiv 1 \mod 8)$ is believed to be either $0$ or $2$. I am ...
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Tamagawa number of elliptic curve of LMFDB
$E_0(\Bbb{Q}_p)=\{P\in E(\Bbb{Q}_p)\mid \tilde{P} \in \tilde{E_{ns}}(\Bbb{F}_p)\}$, here $E_{ns}$ denotes sets of $E(\Bbb{Q}_p)$ which reduces to singular points of $\tilde{E}(\Bbb{F}_p)$.
Thus $E_0(\...
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Connected components of a Shimura variety
Let $(G,X)$ be a Shimura datum in the sense of Definition 5.5 of Introduction to Shimura varieties (Milne, 2017). Let $K$ be a compact open subgroup of $G(A_f)$ that is sufficiently small. Let $S$ be ...
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The relations between two definitions of Hodge bundles
I recently learnt the notion of "Hodge bundle", primarily for families of abelian varieties. This is usually defined as follows: Let $\pi:X\to S$ be an abelian scheme and $e:S\to X$ be its ...
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Group structure of $E(\Bbb{Q})/\hat{\phi}(E'(\Bbb{Q}))$for elliptic curves
Let $E/\Bbb{Q}$ be an elliptic curve and $\phi: E\to E'$ be an isogeny of degree 2 and $\hat{\phi}$ be its dual isogeny.
How can we calculate the group $E(\Bbb{Q})/\hat{\phi}(E'(\Bbb{Q}))$?
In the ...
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kernel of the map $f: E'(K)/\phi(E(K))\to E(K)/2E(K)$
Let $K$ be a number field.
$E/K$ be an elliptic curve over $K$.
Let $\phi : E \to E'$ be an isogeny of degree 2, that is, $\phi・\hat{\phi}=[2]$. Let $\hat{\phi}$ be dual isogeny of $\phi$.
There is a ...
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Is Inverse of bijective morphism also morphism?
Let $V_1,V_2$ be affine varieties.
Let $f$ be a morphism(https://en.wikipedia.org/wiki/Morphism_of_algebraic_varieties) between $V_1,V_2$.
If $f$ is bijective, is inver of $f$ also morphism ?
If not, ...
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Definition of elliptic curve, base point
Let $K$ be a field. Elliptic curve $E/K$ is defined as a genus 1 curve with base point $O$ (for example, in Silverman's book chapter 3).
But in this definition, what is a correct definition of base ...
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Definition of an isogeny of elliptic curves
If we define an isogeny between elliptic curves as a group homomorphism between elliptic curves given by rational polynomial coordinates, what problems arise?
Isogeny is usually defined as a morphism (...
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Basic property and confusion of group law of an elliptic curve
Let $K$ be a number field.
I have a question about elliptic curve $Y^2Z=Z^3-XZ^2\subset \Bbb{P}_K$. I want to know what is the line which go through $[2:\sqrt{6}:1]$(This can be regarded as affine ...
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Group homomorphism of elliptic curve which is not an isogeny
An isogeny $\phi: (E,O)\to (E',O')$ between elliptic curves $E$ and $E'$ is a morphism that satisfies $\phi(O)=O'$.
It is known that $\phi$ is a group homomorphism.
Could you provide an example of a ...
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Adjoint of the derived group
Firstly, let $G$ be a nice linear algebraic group (for instance connected reductive group) over $\mathbb{Q}$. I shall first define the two other groups I require
$\textbf{Definition:}$ If $Z(G)$ is ...
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$j$-invariant of a genus one curve defined over $k$
The theorem and proof are from these notes : https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/a60e5c7aad91910265ee1fc686308413_MIT18_782F13_lec26.pdf
Theorem 26.7.Let $...
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Morphisms to the constant adic space associated to a profinite set
In the Berkeley lecture notes about p-adic geometry, during the fourth lecture, the following claim is made:
Let $S$ be a profinite set and call $\hat{S}$ the affinoid space defined by the ring $C=C^{...
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Curves of genus one,Silverman AEC Exercise 3.22
The question:
Let C be a smooth curve of genus one defined over $K$.
(a)Prove that $j(C)\in K$.
(b)Prove that C is an elliptic curve over $K$ if and only if C$(K)\neq \emptyset$.
(c)Prove that C is ...
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When elliptic curve over local field can be regarded as Tate curve?
I'm completely beginner of Tate curve, sorry to ask a basic question.
I don't see for what condition, elliptic curves can be regarded as Tate curve or not.
For example, let $E_p:y^2=x^3+17x$ be an ...
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Silverman AEC-Theorem 9.3 [duplicate]
Can someone explain why in the last line(image 2),$T\alpha=0\implies T\alpha\beta=0$.
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How to show that Local Neron Height functions are unique
I am reading the proof of Theorem VI.1.1 in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, and I am confused on a step extending a continuous function on $E(K)$ with the $v$-adic ...
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Silverman AEC Theorem 7.4
The theorem:
Let $E_{1},E_{2}$ be elliptic curves and let $\ell\neq$char($K$) be a prime. Then then natural map
$$\operatorname{Hom}(E_{1},E_{2})\otimes \mathbb{Z}_{\ell}\longrightarrow \operatorname{...
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Corollary 6.4:Arithmetic of Elliptic Curves,Silverman
Notation:$\hat{\phi}$ is the dual isogeny for the Frobenius morphism($\phi$).
In proving (c) part of this corollary,we have 2 cases.Either $\hat{\phi}$ is separable or inseparable.Suppose $\hat{\phi}$ ...
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Neukirch 'Algebraic number theory', p73, Prop 11.6. there is a surjection $Cl(O) \to Cl(O(X))$.
This is a question related to Neukirch 'Algebraic number theory',
$p 71$, $Prop 11.6$ (https://web.math.ucsb.edu/~agboola/teaching/2021/fall/225A/neukirch.pdf).
Let $o$ be a Dedekind domain and let $o(...
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Definition of Hodge tensor
The following is the definition of Hodge structures as given in [Milne]:
Let $R$ be one of $\mathbb{R}, \mathbb{Q}$ or $\mathbb{Z}$. And, let $(V,h)$ be an $R$-Hodge structure of weight $n$. Then, ...
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An exact sequence $1 \to {O_{K,S}}^{\times}/{{O_{K,S}}^{\times}}^2 \to K(S,2) \to Cl(K,S)[2] \to 1$
Let $K$ be an imaginary number field.Let $S$ be finite set of places of $K$.
Let $K(S,2)\stackrel{\mathrm{def}}{=} \{b\in K^{\times}/{K^{\times}}^2 \mid v(b)≡0\mod2, \forall v\notin S \}$
Let $S-$ ...
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cardinality of a set in diophantine geometry
Let $a >0$ a real number. I want to compute the cardinality of the set :
$$ \lbrace x \in \mathbb{Q}^{*} \; | \; |x| \leq a \; \text{and } \; v_p(x) \geq 0,\; \forall p \in \mathcal{P} \rbrace $$
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Induced map on 2-part of abelian group
Let $A,B$ be an abelian group. Let $A[2]=\{a\in A\mid 2a=0\}$.
Let $f:A \to B$ be a surjective homomorphism.
Let $f':A[2]\to B[2]$ be induced group homomorphism.
When is $f'$ surjective ?
In the case $...
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Sieves in arithmetic geometry
What are some good resources to learn more about applications of sieve methods in arithmetic geometry?
There are some applications of sieve methods to function fields in the notes of Zeev Rudnick that ...
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Morphisms that are injective and surjective but not isomorphisms
I'm reading Silverman's Arithmetic of Elliptic Curves and read that isogenies are group homomorphisms and if $\phi:E_1\longrightarrow E_2$ is a non-zero isogeny between elliptic curves,its kernel $Ker\...
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Why is $\text{End}(E)\cong\mathbb{Z}[i]$
Silverman in his book Arithmetic of Elliptic Curves has given an example of an elliptic curve $E/K$ with complex multiplication:
$$E: y^2=x^3-x$$
and the map $$[i]:E\longrightarrow E, \ \ \ (x,y)\...
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Contradiction caused by 2-isogeny of elliptic curves?
Let $E:y^2=x^3+x$ be an elliptic curve over $\Bbb{Q}$.
Let $E':y^2=x^3-4x$ be an another elliptic curve.
There is an isogeny $\phi : E\to E'$ given by $(x,y)\to (\frac{y^2}{x^2},\frac{y(1-x^2)}{x^2})...
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Let $E:y^2=x^3+17x$. Let $K=\Bbb{Q}(\sqrt{D})$($D$ is a negative integer). If $D\neq -17$, I want to prove $E(K)_{tor}\cong \Bbb{Z}/2\Bbb{Z}$.
Let $E:y^2=x^3+17x$.
Let $K=\Bbb{Q}(\sqrt{D})$($D$ is a negative integer).
If $D\neq -17$, I want to prove $E(K)_{tor}\cong \Bbb{Z}/2\Bbb{Z}$ admitting the fact that $E(\Bbb{Q})_{tor}\cong \Bbb{Z}/2\...
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Finite subgroups of $\text{PGL}_N(\mathbf{Z})$
Let $p$ be a prime number and $n\ge 1$ an integer. Call $N:=p^n-1$.
Can the cyclic group of order $N$ be contained in $\text{GL}_N(\mathbf{Z})$? Can it be contained in $\text{PGL}_N(\mathbf{Z})$?
...
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What is the definition of $Sha(E/K)[\phi]$ for elliptic curves?
Let $\phi: E \to E'$ be isogeny between elliptic curves.
Let $K$ be a number field. Let $Sha(E/K)$ be a Tate-Shafarevich group of elliptic curve $E/K$.What is the definition of $Sha(E/K)[\phi]$ ?
...
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degree 2 isogeny $\phi : E\to E'$ given by $(x,y)\to (\frac{y^2}{x^2},\frac{y(a-x^2)}{x^2})$ and $\phi(0,0)$
Let $K$ be a field of characteristic $0$. Let $a\in K$.
Let $E:y^2=x^3+ax$ and $E': y^2=x^3-4ax$.
There is an degree 2 isogeny $\phi : E\to E'$ given by $(x,y)\to (\frac{y^2}{x^2},\frac{y(a-x^2)}{x^2})...
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Silverman Elliptic Curves Exercise 1.8
I am reading Silverman's Arithmetic of Elliptic Curves.I was trying to solve exercise 1.8 but I'm stuck.
1.8. Let $\mathbb{F}_q$ be a finite field with q elements and let $V\subset \mathbb{P}^{n}$ be ...
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Rational maps from $\mathbb{P}^{1}$ to $\mathbb{P}^{1}$
I was reading Arithmetic of Elliptic Curves by Silverman,I had the following question:
Is there a way to classify rational maps from $\mathbb{P}^{1}$ to $\mathbb{P}^{1}$ over some algebraically closed ...
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When are mirror-image modular curves isomorphic to each other?
Let $R$ be the matrix $\left[\begin{smallmatrix}-1 & 0\\ 0 & 1\end{smallmatrix}\right]$. Let $s$ be the involution of $SL_2(\mathbb{Z})$ given by $s(A) = RAR$
Let $\mathbb{H}$ be the upper ...
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Existence of non-degenerate alternating pairing $ \psi : A \times A \to \mathbb{Q}/\mathbb{Z} $ and perfect square cardinality
Let $A$ be a finite abelian group, and let
$ \psi : A \times A \to \mathbb{Q}/\mathbb{Z} $
be an alternating, non-degenerate bilinear form on $A$.
To prove $A$ have square cardinality, Non-degenerate ...
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Proper scheme over the ring of integers of a local field
I saw such a fact in a paper but I don't know why it's true. Let $K/k_\infty$ be a finite extension, where $k$ is a global function field and $\infty$ is a place. Let $\mathcal{O}_K$ be its ring of ...
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