All Questions
30
questions
8
votes
0
answers
290
views
Do automorphisms actually prevent the formation of fine moduli spaces?
I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...
- 821
1
vote
0
answers
99
views
Compactifications of product of universal elliptic curves
Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving ...
- 12k
6
votes
1
answer
284
views
Definition of modular curve associated to $\Gamma(N)$
For a positive integer $N$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{...
- 821
3
votes
1
answer
389
views
Moduli space of genus 1 curves with a degree n divisors
I am sure this is well known, but I don't know what to search for:
Consider $M_{1,n}$, the moduli space of genus 1 curves with $n$ marked points. The symmetric group on $n$ letters acts on this space ...
- 7,716
8
votes
0
answers
171
views
Geometry of moduli problem in practice: how to check it is connected / irreducible / normal / reduced / locally complete interesection...?
Moduli spaces are very common and useful in the world of algebraic geometry. From the point view of functors, one can already check many geoemtric properties of it. I like examples, and you can assume ...
- 461
6
votes
1
answer
651
views
On the moduli stack of abelian varieties without polarization
(I am especially interested in abelian surfaces and characteristic 0).
How bad is the moduli stack of abelian varieties (with no polarization or level structure)? Is it an Artin stack? DM (Deligne-...
- 7,716
8
votes
0
answers
388
views
Stacky proof of no elliptic curves over Z
It is a well known result that there are no Elliptic curves over the integers with every where good reduction. In fact this is even true for abelian varieties (and hence higher genus curves) but let ...
- 7,716
6
votes
0
answers
163
views
What are the genus 4 curves with Jacobians that are 4-th powers?
Consider the moduli space of all genus $4$ curves $\overline{\mathscr M_4}$ of dimension $3\times 4 - 3 = 9$. Under the Torelli map, there is a map to $\overline{\mathscr A_4}$ (which has dimension $...
- 7,716
4
votes
0
answers
188
views
lemma II.2.4 in Harris-Taylor (about drinfeld-katz-mazur level structure on 1-dimensional $p$-divisible groups)
Lemma II.2.4 on page 82 in Harris and Taylor's "The Geometry and Cohomology of Some Simple Shimura Varieties" (or lemma 3.2 here), says that given a Drinfeld(-Katz-Mazur) level structure $\alpha:(p^{-...
- 371
16
votes
2
answers
2k
views
Good introductory references on moduli (stacks), for arithmetic objects
I've studied some fundation of algebraic geometry, such as Hartshorne's "Algebraic Geometry", Liu's "Algebraic Geometry and Arithmetic Curves", Silverman's "The Arithmetic of Elliptic Curves", and ...
- 1,352
2
votes
1
answer
477
views
Does Lang's conjecture imply Morton-Silverman's Uniform Boundedness conjecture?
I was curious to see whether the following conjecture of Morton-Silverman is (known to be) a consequence of Lang (or Lang-Vojta's) conjecture.
Conjecture. Let $D$, $N$, and $d$ be positive integers. ...
- 9,447
3
votes
0
answers
132
views
Arithmetic version of "Attaching maps" for moduli of curves
I am looking for a reference for attaching maps of moduli of curves with marked points. Especially I would like to know whether they descend over $\mathbb{Z}$. On one hand this seems very hard to ...
- 845
1
vote
0
answers
132
views
Is there a reference for boundedness of smooth canonically polarized varieties over Z (No...)
In Kollár's paper Quotient spaces modulo algebraic groups, Kollár mentions right above Theorem 1.8 that the stack $\mathcal M_P$ of smooth canonically polarized varieties over Spec $\mathbb Z$ with ...
- 11
5
votes
1
answer
321
views
Regular minimal model of $X_0(p^2)$
Consider the compactified modular curve $X_0(p^2)$ and the corresponding algebraic curve over $\mathbb{Q}$. My questions are the following:
Where do the cusps of $X_0(p^2)_{\mathbb{Q}}$ live? That is ...
- 617
11
votes
2
answers
613
views
Easiest example where field of definition is not field of moduli
There are many examples of varieties over $\overline{\mathbb Q}$ whose field of moduli is $\mathbb Q$ but which can't be defined over $\mathbb Q$. What is the easiest such example? It should be a ...
- 111