S_hilbert_symbol() helper function to find integer with negative Hilbert symbol for primes in a set S

[Anna Haensch]

This is an implementation of algorithm 3.4.1 from Markus Kirschmer's "Definite quadratic and hermitian forms with small class number." Given a non-square b and a set of primes S, the function returns a number a which has negative Hilbert symbol with respect to b at precisely the primes in S.

This is a function in service to algorithm 3.4.3, quadratic_form_form_local_invariants(), ticket #24108.

[Anna Haensch]

New commits:

​ec55c0d

Adds function S_hilbert_symbol

[Simon Brandhorst]

So here are some thoughts:

This is just a helper function so it should not pollute the global namespace. Is there a reason that you have implemented Markus algorithm just for QQ? The beauty of Markus' work is that he did it directly for number fields. It should not be too hard to do it for number fields. All the necessary functions like class groups, ideals, hilbert symbols, places, are there:

sage: K.<a>=CyclotomicField(13)
sage: P=K.primes_above(3)[0]
sage: P
Fractional ideal (3, a^3 - a - 1)
sage: K.hilbert_symbol(3,1+a,P)
1

We can define the function as a method of the number field. Just like the hilbert_symbol. Maybe the name S_hilbert_symbol is misleading? I would expect it to be the global hilbert_symbol of the localization at S or something like that. This function seems more like an inverse image? maybe hilbert_symbol_inv_S?

Some remarks for formatting: Comments should be of the form (note the space)

# foo does bla
foo = 3 

See ​https://www.python.org/dev/peps/pep-0008/#comments.

You are redefining V,W,vv in the while loop. But really they stay constant. So they should not sit in the loop.

Use the names of Markus if possible or more descriptive names. HH,PP etc. is not very descriptive. Same for i,j. I do not see the need to define H. Why hide everything in lists?

Either use the verbose things or remove them.

[Simon Brandhorst]

It is good to test the error messages. But they belong to the TESTS:: block not the EXAMPLES:: block. -> they do not need to show up in the documentation.

[Simon Brandhorst]

How about hilbert_conductor_inverse as a name? Problem is this exists and is (allmost) the same function. Ideas? I would like to move it as a method of QQ.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​61faa15

Clean up

[Anna Haensch]

Replying to sbrandhorst:

How about hilbert_conductor_inverse as a name?

I'm not sure about that name, it already exists and means something else. I am happy with S_hilbert_symbol since it's restricting the Hilbert symbol at a finite set of primes, it's pretty consistent with your initial interpretation in comment 9; a hilbert symbol taken at a finite number of localizations.

Problem is this exists and is (allmost) the same function. Ideas? I would like to move it as a method of QQ.

Eventually we can split this into two methods, one for QQ an one for number fields. I've got the general number fields method queued up and ready to go as soon as tickets #25023 and #25029 are positively reviewed.

[Simon Brandhorst]

Mhh I do not like the prefix S_. In methods like S_unit_group and S_class_group, the primes in S are inverted. But we are not inverting anything - we are working in fields anyways. So there is no relation to this kind of objects? Finally in these S-rings one can define a hilbert symbol and we are not computing it.

For b and S given our function computes a such that hilbert_conductor(a,b) = prod(S) plus a condition at infinity. So I think that it is pretty close. And I like the idea that if you type hilbert + autocompletion you get this method suggested. So maybe

• hilbert_something
• hilbert_conductor_something
• hilbert_conductor_inverse with optional argument b?
• hilbert_conductor_inverse_S?

ooops I have the numberfield version ready as well

[Anna Haensch]

How about hilbert_symbol_positive_at_S?

[Simon Brandhorst]

i could settle for hilbert_symbol_negative_at_S

:)

[Anna Haensch]

Replying to sbrandhorst:

i could settle for hilbert_symbol_negative_at_S

:)

Ah, yes, even better. Let's go with that :)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​7077033

changed name to hilbert_symbol_negative_at_S

[Simon Brandhorst]

Would you review the number field version? Else we can split it off to another ticket.

[Anna Haensch]

This ticket has been asleep for awhile, but let's try and finish it. I just tried to run doctests on and got an error message

AttributeError: Values instance has no attribute 'gc'

Any ideas?

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​9e5a98f	Merge branch 'develop' into t/24098/_s_hilbert_symbol___helper_function_to_find_integer_with_negative_hilbert_symbol_for_primes_in_at_set_s
​b7ca8fe	fixed doctests
​1fbeefc	Merge branch 'develop' into t/24098/_s_hilbert_symbol___helper_function_to_find_integer_with_negative_hilbert_symbol_for_primes_in_at_set_s

[Simon Brandhorst]

should be okay now.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​1a6e6da

removed unnecessary import

[Simon Brandhorst]

We have a green bot. Tests pass.

[Anna Haensch]

Replying to sbrandhorst:

We have a green bot. Tests pass.

All tests pass! I had just run them independently anyhow, but good to know :)

[Anna Haensch]

I fixed a few errors that I spotted in the reference manual build, everything else looks good. What do you think?

---

New commits:

​6b72793	fixed errors in reference build
​ca2db75	fixed typo in documentation and code commenting
​bca5e62	fixed typo again

[Simon Brandhorst]

Add your name as a reviewer, then we can set positive review.