The function completions applied to a modular symbol taking values in Sym^k(K) where L is a number field returns a list of pairs: (modular symbols with values in Q_p, map from K --> Q_p) except in the case when K admits no maps to Q_p. In this case, the function takes the defining polynomial of K, say f, and tries to form:
L = Qp(p,M).extension(f)
-- which currently only works if f is unramified or eisenstein -- and then returns the ordered pair: (mod sym with values in L, map form K --> L).
Questions:
1) Should this function return the list of length 1 with the above data (instead of just an ordered pair)?
2) Why not return the list of all completions of primes over p? I don't understand how the map from K to L is chosen above. What if K is quintic over Q, and it's defining polynomial factors into a quadratic and a cubic over Q_p. How do you know which map to pick going from K to L?
The function completions applied to a modular symbol taking values in Sym^k(K) where L is a number field returns a list of pairs: (modular symbols with values in Q_p, map from K --> Q_p) except in the case when K admits no maps to Q_p. In this case, the function takes the defining polynomial of K, say f, and tries to form:
L = Qp(p,M).extension(f)
-- which currently only works if f is unramified or eisenstein -- and then returns the ordered pair: (mod sym with values in L, map form K --> L).
Questions:
1) Should this function return the list of length 1 with the above data (instead of just an ordered pair)?
2) Why not return the list of all completions of primes over p? I don't understand how the map from K to L is chosen above. What if K is quintic over Q, and it's defining polynomial factors into a quadratic and a cubic over Q_p. How do you know which map to pick going from K to L?