The two definitions differ when there are primes of K dividing m that don't lie in S.
My personal preference is for the "unramified outside S" definition, because it seems natural in all contexts. I can define local conditions for a generalized Selmer group consistent with your definition, but they're a little unnatural. On the other hand, can see wanting one definition today, and the tomorrow. In the event of a draw, I prefer to err on the side of picking the larger group (your definition) since it's less likely to result large scale programs spitting something out that's false. Typical results about Selmer groups involve obtaining upper bounds on such groups, or enumerating things that will be eliminated later in the program. Your definition also preserves the exactness of the diagrams on the backs of certain green shirts. Jamie Weigandt On Jun 10, 2014, at 11:59 AM, John Cremona <[email protected]> wrote: > I was intending to extend the function K.selmer_group(S,m) so that it > could used for K=QQ without having treat this as a special case in my > code, or use a trick such as defining Q=NumberField(x-1): > > > sage: Q.<t>=NumberField(x-1) > sage: Q.selmer_group(Q.primes_above(7),2) > [7, -1] > sage: Q.selmer_group(Q.primes_above(5),2) > [5, -1] > > This output agrees with my definition of the K-selmer-group usually > denoted K(S,m), as the subgroup of K^*/(K^*)^m whose representative > elements a in K^* satisfy m | ord_p(a) for all p not in S. But the > docstring for the selmer_group function claims something slightly > different: > > Compute the Selmer group `K(S,m)`. This is the subgroup of > `K^\times/(K^\times)^m` consisting of elements `a` such that > `K(\sqrt[m]{a})/K` is unramified at all primes of `K` outside > of `S`. > > (The function returns generators of the group rather than the group > itself, but I don't mind that for now and tha is not what I am talking > about here.) > > According to this, QQ([5],2) should be [5] and QQ([7],2) should be > [-7], I think. And in general for m=2, the output should have a > generator +p for primes p in S which are 1 (mod 4) and -p when p=3 > (mod 4), and both -1 and +2 when 2 is in S. > > I suggest that, rather than change the function to do what it says it > does (which would be rather delicate for a general number field and > general m), that the docstring is changed to something like "my" > definition above. Do people agree? Then the above output is correct > and I can continue to implement the version for QQ itself. > > John > > -- > You received this message because you are subscribed to the Google Groups > "sage-nt" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send an email to [email protected]. > Visit this group at http://groups.google.com/group/sage-nt. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-nt" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send an email to [email protected]. Visit this group at http://groups.google.com/group/sage-nt. For more options, visit https://groups.google.com/d/optout.
