8
votes
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
8
votes
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
4
votes
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
2
votes
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
2
votes
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
2
votes
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
2
votes
Computing Thompson series for the monster group
In the OEIS index, see
"McKay-Thompson sequences or series for Monster simple group, sequences related to"
These may come with tables of coefficients, and with programs to compute ...
- 37.2k
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