Reducing Gamma(N) cusps does not always reduce

[JohnCremona]

Steps To Reproduce

In Sage-10.1:

sage: Gamma(7).reduce_cusp(Cusp(6,7))
6/7
sage: Gamma(7).reduce_cusp(Cusp(1,0))
Infinity
sage: Gamma(7).are_equivalent(Cusp(1,0), Cusp(6,7))
True

Expected Behavior

Both 6/7 and 1/0 should reduce to Infinity = 1/0 since 6/7 and 1/0 are equivalent modulo Gamma(7), and according to the definition of a reduced cusp for Gamma(N). So we should see

sage: Gamma(7).reduce_cusp(Cusp(6,7))
Infinity

Actual Behavior

instead of

sage: Gamma(7).reduce_cusp(Cusp(6,7))
6/7

Additional Information

The code reduces both u and v mod N (where the cusp is u/v and the level is N) and then calls the hidden function _lift_pair(u,v,N). But that function assumes that its input u,v are coprime, which in this case they are not.

To fix this it suffices to change _lift_pair() to return Cusp(1,0), i.e. Infinity, when v%N==0 and u%N is either +1 or N-1; the second case was being missed.

Environment

- **OS**: Ubuntu 22.04.3 LTS
- **Sage Version**: 10.1

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide