Author: William A. Stein
Component groups of J0(N)(R) and J1(N)(R)

# Component Groups of J0(N)(R) and J1(N)(R)

## J0(N)(R)

I computed the component groups for N<=500, and here is the data as a list of pairs N, order of component group at infinity.

The first example with nontrivial component group is the elliptic curve J_0(15).

 Order Number of N<=500 with component group of this order 1 212 2 253 4 0 8 35 >8 0

This data is intriguing and suggests something interesting is going on. Why doesn't 4 ever occur as the order of a component group? Why nothing bigger than 8 (up to level 500)? Very weird. I bet there's a theorem here waiting to be proved.

## J1(N)(R)

I computed the component groups for N<=500, and here is the data as a list of pairs N, order of component group at infinity.

For J_1(N) the data is much harder to compute. I computed it for levels N <= 54
and found the following:

 Order Number of N<=54 with component group of this order 1 37 2 8 4 4 8 3 16 1 32 1 >32 0

Here component groups of order 4 do occur, and even one as big as 32.