# Tagged Questions

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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### Multiplication by $m$ isogenies of elliptic curves in characteristic $p$

I've been attempting to prove some comments I've read on MO by myself for my undergrad thesis regarding étale morphisms of elliptic curves. My definition of an étale morphism is taken from Milne's ...
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### Geometrically ireducible curve

I know that curve with coefficients in $k$ is geometrically ireducible if it does not factor over algebraic closure of $k$. I have this curve, for example, $$2x^2+2x^2y+2y^2+2xy+3xy^2=1.$$ It's ...
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### J. Silverman exercise 3.12 “The arithmetic of Elliptic curves”

I have question regarding exercise 3.12 of J. Silverman "The arithmetic of Elliptic curves". It states the following: Let $m \geq 2$ be an integer, prime to $\text{char}(K) > 0$. Prove that the ...
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### Generalized elliptic curves over cusps and orbits of $\mathbb{Q}\cup\infty$

In the post http://mathoverflow.net/questions/51147/what-objects-do-the-cusps-of-modular-curve-classify, it says that the fibers over the cusps of a modular curve are n-gons. Wikipedia (https://en.m....
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### Tate curve and cusps

I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve? Naive googling seems to suggest that level structures on the Tate curve (up to ...
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### Let $K = \mathbb{F}_p(t)$, $X$ be the affine curve $(t^5 + 1)y^2 = x^5 + 1$, what is $p_g(X)$? Is $X(K)$ finite? [closed]

Let $p$ be a prime not equal to $2$ or $5$. Let $K = \mathbb{F}_p(t)$. Let $X$ be the affine curve $(t^5 + 1)y^2 = x^5 + 1$ in $\mathbb{A}_K^2$. I have two questions. What is $p_g(X)$? Is ...
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### $E[n]$ is etale locally $(\mathbb{Z}/n\mathbb{Z})^2$

I don't think we need the entire setup below (from Katz and Mazur's elliptic curve book, pages 74 and 75), but, as a beginner, I am unable to identify the assumptions I need. Let $S$ be the open ...
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### descending smoothness along unramified morphisms

Let $f:X\rightarrow Y$, and $g:Y\rightarrow Z$ be morphisms of schemes locally of finite type. Suppose that $f$ is unramified and surjective, $g$ is flat and $g\circ f$ is étale. Does $g$ is also ...
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### Elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic

I am finding elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic. Elliptic curves over $\mathbb C$ can be identified with $\mathbb C/\land$ for some lattice $\land$. And ...
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### Asking for some exercises to help me understanding abelian varieties better?

I want to study Mumford's Abelian Varieties in the coming winter break. I tried to study it before, but I didn't find my self really understanding(or memorizing) too much. I guess a better and more ...
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### Weil pairing of curve of genus 2

We know there is Weil pairing for elliptic curve satisfying several nice properties. So do we have Weil pairing for other curves also satisfying the nice property? Especially genus 2 curve?
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### X a genus 2 curve. Element of prime order in Aut(X) has order at most 5.

Let X a genus 2 curve. Aut(X) finite. I want to prove that Element of prime order in Aut(X) has order at most 5. What I know is non-identity endomorphism of X acts nontrivially on the Tate module of ...
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### 2-torsion points in a curve with genus 2

Let X be a genus 2 curve with affine equation y^2 = f(x), where f is a polynomial of degree 6. Write $P_1, ..., P_6$ for the points on X(C) with y=0. Then why every $P_i-P_j$ is a 2-torsion points in ...
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### Quotienting out polynomial rings by polynomial-generated ideals

I'm studying in Lorenzini's Arithmetic Geometry, and it has been a while since I've taken a rigorous algebra course. I'm trying to understand a certain step in his proof that $\mathbb{Z}[\sqrt{5}]$ ...
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### Smooth curve defined over K with genus 1 is always isomorphic, over $\overline K$, to an elliptic curve over K

Here, the point is the smooth curve defined over K with genus 1 may not have rational point. But to be an elliptic curve defined over K, the base point must be a rational point. I tried to use ...
For $m\ne2$ I want to show that if two automorphisms coincide on $E(m)$, which is the $m$-torsion subgroup of the elliptic curve $E$, then these automorphisms are the same. The statement is very ...
Let $X$ be a scheme of finite type over a field $k$ which is algebraically closed of characteristic zero. Let $K$ be another field and $\eta \in X(K)$ be a $\mathrm{Spec}(K)$ point of $X$. Is the ...