Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn’t find it in Diamond and Shurman, and neither could I find an explicit formula with a simple google search. Certainly, there should be some explicit expression, no?

In this paper by Minhyong Kim on p5, there is a variety $X$ defined over $\mathbb{Q}$, $G = \pi_1(X(\mathbb{C}),b)$ the topological fundamental group of the associated complex algebraic variety, and $G$^ the profinite completion of $G$. Kim states that $G$^ is a sheaf of groups for the etale topology on Spec($\mathbb{Q}$). Why is this? A […]

How to compute the genus of $ \{X^4+Y^4+Z^4=0\} \cap \{X^3+Y^3+(Z-tW)^3=0\} \subset \mathbb{P}^3$? We know that the genus of $ \{X^4+Y^4+Z^4=0\} \subset \mathbb{P}^3$ is 3 because the degree is 4. Now, I want to know the genus of the intersection as a curve. For that I have to use the adjunction formula and the fact that […]

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from algebraic geometry. I don’t want a too rigorous approach, made of a lot of definition and propositions but instead […]

This is question 7.2.3 in Liu’s book Algebraic Geometry and Arithmetic Curves and I have been trying with this for some time now. Let $f:X \rightarrow Y$ be a morphism of Noetherian schemes, and suppose that X and Y are integral and that f is finite surjective. We will let $Div(Y)$ resp. $Div(X)$ stand for […]

Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive integer values of $t$ the respective specialization has a point over $K$? (I expect the answer to be […]

Why a smooth surjective morphism of schemes admits a section etale-locally?

According to Bjorn Poonen’s notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective algebraic curves. This suggests that the archimedean places are infinite in the sense of being ‘points at infinity’ and not just because rational integers […]

I’ve been reading through Neukirch’s Algebraic Number Theory, and I’m a little puzzled about a possibility with ramification of primes. As usual, let $\mathcal{O}_K$ be a Dedekind domain with field of fractions $K$, let $L/K$ be a finite algebraic extension, and let $\mathcal{O}_L$ be the algebraic closure of $\mathcal{O}_K$ in $L$. Let $\mathfrak{p}$ be a […]

The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. If $K/\mathbf Q$ is a number field, Dirichlet’s theorem says (among other things) that the group of units $\mathcal O_K^\times$ is finitely generated. The proof of Mordell’s theorem and the proof of Dirichlet’s theorem […]

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Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible