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I have come across the following surface: let $X$ be the double covering of $\mathbb{P}_\mathbb{Z}^2$ defined by the equation $$y^2=x_0^6+x_1^6+x_2^6$$ where $y$ is a variable of degree 3. There is a …

Given an arithmetic surface $S$, I would like to know the following properties of its first and second Chow groups $CH^1(S), CH^2(S)$: Are they finitely generated? If so, what is the rank? What is t …

Given a cubic number field and a basis $\{\gamma_1,\gamma_2,\gamma_3\}$ for it over the rationals, we can write down the norm equation $N(x_1\gamma_1+x_2\gamma_2+x_3\gamma_3)=1$. For almost all substi …

This is a question on math.se that got no answers. 1) Is there a relatively general method of computing the dimensions of representations in a reducible induced representation? An explicit specific …

Of course not. But after reading a bit, some points make me believe it should be: Let $S$ be a nice$^{\*}$ surface defined over $Spec\ \mathbb{Z}$. The Brauer group $Br(S\otimes \bar{\mathbb{Q}})$ …

Let $E_{ij}(x)\in \mathrm{Mat}_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $$a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x),\quad b(y)=1+E_{23 …

I have been trying to reconstruct some elliptic curves theory computationally and have gotten stuck on some period computations. Specifically, let $$E_\lambda:\ y^2=x(x-1)(x-\lambda)$$ be the Legendr …

The primes that are "bad" in your sense will divide the number $Res_x(Res_y(f,\frac{\partial f}{\partial x}), Res_y(f,\frac{\partial f}{\partial y}))$. (If I interpreted damiano's comment correctly). …

Background We are given a curve with integer coefficients. I want to make a suggestion in another question (Computationally bounding a curve's genus from below?) into a deterministic algorithm for fi …

Given a CM field we can use its maximal order (and a choice of CM type) to construct an abelian variety $\mathbb{C}^g/\Lambda$ with complex multiplication by the maximal order. How do I (or where can …

Background In the course of answering another question (Infinite collection of elements of a number field with very similar annihilating polynomials) I found myself with a curve, that if it had a pos …

If doing geometry over $\mathbb F_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of $\mathbb F_1$ - the field with one element. I saw that the f …

David's question Families of genus 2 curves with positive rank jacobians reminded me of a question that once very much interested me: when is a product of elliptic curves isogenous to the jacobian of …

In a discussion today on the Shafarevich-Tate group of an elliptic curve, the following structure and question came up. I will abuse many notations and be very vague about some things, but am very ope …

Question Given an abelian variety $V$ and an integer $n$, is there a natural abelian category with a natural object $X$ and natural coefficients $F$ so that $V\simeq H^n (X,F)$? Motivation Studying …