57
votes

Accepted

### Clausen's modified Hodge Conjecture

It's a bit of a long story, but I can at least give the idea. Let $X$ be a smooth projective variety over $\mathbb{C}$. The Hodge conjecture says that for all $p\geq 0$, the cycle class map
$$Ch^p(X)...

15
votes

### Does this conic have a rational point?

The answer is no. If there was, we could assume that $X,Y,Z$ are in $\mathbb Q[u,v]$ and are coprime (since that ring is a UFD). Setting $v=0$ we get $X(u,0)^2+uY(u,0)^2=0$ in $\mathbb Q[u]$, which (...

12
votes

Accepted

### Definition of locally symmetric space of reductive groups

There is a very natural, intrinsic definition of a "symmetric space", as a manifold (Riemannian or Hermitian) with an extra symmetry of a certain prescribed type. It is then a theorem, not a ...

10
votes

Accepted

### Definition of modular curve associated to $\Gamma(N)$

This is a subtle issue (which has come up before on this site several times, see e.g. is the modular curve X(N) defined over Q? for a related question).
Your $S(N)$ is naturally a scheme over $\mathbb{...

8
votes

Accepted

### Étale group schemes and specialization

If $k \to \ell$ is any regular field extension (i.e. $k$ is algebraically closed in $\ell$), then $X(k) \to X(\ell)$ is a bijection when $X \to \operatorname{Spec} k$ is étale. Indeed, it suffices to ...

6
votes

Accepted

### Is the value of the power series at 0.1 transcendental?

This question is likely open.
We can tell whether $f(1/10)$ is rational or irrational (by asking whether $a_n$ is eventually periodic); in this case, definitely irrational.
Can we have an algebraic ...

6
votes

Accepted

### Faithful representations of integral models

Yes, there exists a closed immersion $\mathcal{G}\to \mathrm{GL}_n$ over
$\mathbb{Z}$. This is folklore. For a proof, see for example Proposition 3 of arXiv:2012.05708v3

5
votes

### The notion of morphisms between two moduli problems in Katz-Mazur

That's correct. We would like this to give a morphism between the underlying stacks, i.e. the functor that takes $S$ to the groupoid of pairs of an elliptic curve $E/S$ and an element of $\mathcal P_i(...

4
votes

### Is there a method to check if two sections of an elliptic surface are dependent over the endomorphism ring or not?

Let $\mathcal O=\mathbb Z[\tau]=\mathbb Z+\mathbb Z\tau$. Then you're asking if the four points
$$
P_0(t),\; \tau\bigl(P_0(t)\bigr),\; P_1(t),\; \tau\bigl(P_1(t)\bigr)
$$
are $\mathbb Z$-linearly ...

4
votes

Accepted

### Flat scheme-theoretic closure

Explicitly, lets let $R = \mathbb{C}[x]_{(x)}$ so $K = \mathbb{C}(x)$. You can then let $C_K = {\bf P}^1_K$ and $C_R = {\bf P}^1_R$. $C_R$ has a chart that looks like $\mathbb{C}[x]_(x)[y]$. This ...

4
votes

### Zariski dense in abelian scheme

How about if $A = E_1\times E_2$ with $E_i$ non isotrivial elliptic curves and $s: S \to P\times 0$. Then your $\mathbb Z\cdot X$ is contained in $E_1\times 0$ and so is not Zariski dense. In fact, ...

4
votes

### Rational points on genus 3 curves defined by short equations

I have looked a bit at the first equation. It has (at least) seven rational points (as a projective curve). The differences of these points generate a free abelian group of rank three in the Mordell-...

2
votes

### Is the value of the power series at 0.1 transcendental?

For the UPDATE, allowing coefficients $a_n < M$ for a fixed $M$. Then there are examples with $f(1/10)$ rational.
Let's do this. Define a sequence $(a_n)$ of coefficients as follows: Start with ...

1
vote

### Symmetric powers, localisation and Frobenius

I want to come back to this, and provide a sketch of the Dold-Puppe argument so people who don't speak German (such as myself) don't have to try and sift through the linked paper to try and find the ...

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