# Tag Info

Accepted

### Non-modular elliptic curves

It is a widely believed conjecture that all elliptic curves, over any number field $K$, are modular (in the sense that there exists an automorphic representation [*] $\pi$ of $\operatorname{GL}_2 / K$ ...
• 31.5k
Accepted

### Fricke involution and Atkin operator

EDIT. In the answer below, $U_q$ refers to the usual Hecke operator given on Fourier expansions by $\sum_{n \geq 1} a_n x^n \mapsto \sum_{n \geq 1} a_{qn} x^n$. The operator $U_q$ in the OP is given ...
• 19.7k
Accepted

### Reference for universal elliptic curves

For any $n\geq 1$, one can define a functor $\mathcal{M}_1(n)\colon \mathrm{Schemes}/\mathbb{Z}[\frac1n] \to \mathrm{Groupoids}$, sending a scheme to the groupoid of elliptic curves over it with a ...
• 11.4k

### Reference for universal elliptic curves

To complete Lennart Meier's nice answer, Baaziz has computed explicit equations for the universal elliptic curve over $Y_1(N)$ up to $N=51$. The method used and the data up to $N=20$ can be found in ...
• 19.7k
Accepted

### Overconvergent modular forms and the level at $p$

The curve $X_1(Np^n)$ is connected, but the ordinary locus in this curve is not: if you remove the residue discs of the supersingular points, what's left "falls apart" into a disjoint union ...
• 31.5k
Accepted

### Non-vanishing modular forms

I will answer Q2: N=2: Denote by $Y^1(2)$ the moduli of elliptic curves with point of order 2 and fixed invariant differential. It is not hard to show that \$Y^1(2) = \mathrm{Spec}\, \mathbb{Z}[\frac12]...
• 11.4k