# 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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### 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, $\phi・\hat{\phi}=$. 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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### 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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### 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 $\text{PGL}_N(\mathbf{Z})$? ... ### 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 $Sha(E/K)[\phi]$ ? ...