3
votes
0answers
122 views

Moduli interpretation of Eisenstein series

Let $N \geq 11$ be an integer and consider the basis of Eisenstein series for $M_2(\Gamma_0(N))$ described in Theorem $4.6.2$ of Diamond--Shurman's book. Pick and Eisenstein series $F$ in this basis. ...
4
votes
0answers
119 views

Moduli interpretation of Hecke operators on Shimura curves

In his book on Automorphic Forms, Shimura gives (chapter 9) definitions of the the Hecke operators for Shimura curves. One can give definitions of the Hecke and Atkin-Lehner operators in terms of the ...
5
votes
2answers
204 views

Why is the supersingular locus the zero locus of a modular form?

This question is related to my other question here: Examples of subspaces singled out by modular forms. Here I am wondering if there is a philosophical explanation about why the supersingular locus ...
2
votes
0answers
79 views

Examples of subspaces singled out by modular forms

I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature. To be more clear let me explain the example I have in mind. Let ...
11
votes
0answers
188 views

What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?

I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that ...
3
votes
1answer
136 views

On Universal Abelian surfaces over a Shimura curve.

Let ${\cal O}, {\cal O}'$ be two order in ${\mathrm M}_2({\Bbb R})$ that are sets of all $2 \times 2$ matrices over real number ${\Bbb R}$. Assume that we have the relation ${\cal O}' = a{\cal ...
10
votes
0answers
359 views

Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties. Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac ...
0
votes
0answers
117 views

Action on C[[X,Y]]/f(X,Y) giving complete intersection quotients

Let $R \colon\!= {\Bbb C}[[X,Y]]/(f(X,Y))$ be a complete local ring of Krull-dimension $1$. Assume that we have an action of $\Bbb Z$ on $R$ such that fixed elements by $\Bbb Z$ in $R$ are only ...
9
votes
1answer
327 views

examples of “exotic” moduli problems for elliptic curves?

Let $\textbf{Ell}$ be the category of elliptic curves over various base schemes, and where a morphism between $E\rightarrow S$ and $E'\rightarrow S'$ is a cartesian diagram with those two maps as ...
1
vote
1answer
120 views

reference request for the finiteness of cuspidal subgroup of $X_0(N)$?

I've seen stated offhand in many sources that the cuspidal subgroup of the Jacobian of $X_0(N)$ is finite. Do they mean that the subgroup of the jacobian generated by $\mathbb{Q}$-rational cusps is ...
8
votes
1answer
533 views

what exactly is the moduli functor for classifying elliptic curves with (full) level N structure?

So, when people say, "the moduli problem of classifying elliptic curves over $\mathbb{C}$ with level $N$ structure", there are usually two associated functors I've seen: $P_N : ...
5
votes
2answers
557 views

moduli interpretations for modular curves

Some big picture questions - What are some applications of the moduli interpretation for congruence curves? Specifically, the interpretations for congruence curves parametrizing elliptic curves with ...
5
votes
1answer
426 views

Some help in digesting a paragraph in the introduction of Deligne/Rapoport's “Les Schemas de Modules de Courbes Elliptique”

http://www.springerlink.com/content/04x54gr171v556m4/fulltext.pdf On page 149 (DeRa-7), in the middle of the page, I can translate the middle paragraph that starts "3. La surface de Riemann ..." as ...
2
votes
1answer
403 views

When does a Shimura variety have contractible universal cover?

Disclaimer: I know very little about Shimura varieties. Some Shimura varieties have a contractible universal covering space, for instance $A_g$ itself. Are there any nice necessary and/or sufficient ...
3
votes
0answers
545 views

level structures and moduli of abelian varieties

Hello, In the definition of level structure of level $n$ for an elliptic curve $A$, there are two versions: an isomorphism of group schemes $(\mathbf Z/n\mathbf Z)^2 \to A[n]$. an isomorhpism of ...
6
votes
3answers
640 views

Upper bounds for ranks of modular jacobians

The following question came to me earlier as a "side question"; something I'd like to know, but which is not totally necessary for what I'm thinking about or doing: Consider the genus 32 curve ...
4
votes
2answers
1k views

(nontrivial) isotrivial family of elliptic curves

I think it should be a standard procedure to construct such things, can anyone give a reference or give a hint? Can this be done over any base scheme?
3
votes
1answer
1k views

moduli space and modularity

I recently realized some kind of analogy when considering modularity results (such as the modularity of elliptic curves over Q). The analogy comes from algebraic groups. Take one point (say, the ...
11
votes
2answers
1k views

Why is one interested in the mod p reduction of modular curves and Shimura varieties?

Why is one interested in the mod p reduction of modular curves and Shimura varieties? From an article I learned that this can be used to prove the Eichler-Shimura relation which in turn proves the ...
8
votes
4answers
972 views

cohomology of moduli spaces

Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...