def padic_gauss_sum(a,
p,
f,
prec=20,
factored=False,
algorithm='pari',
parent=None):
# Copied from Sage
from sage.rings.padics.factory import Zp
from sage.rings.all import PolynomialRing
q = p**f
a = a % (q - 1)
if parent is None:
R = Zp(p, prec)
else:
R = parent
out = -R.one()
if a != 0:
t = R(1 / (q - 1))
for i in range(f):
out *= (a * t).gamma(algorithm)
a = (a * p) % (q - 1)
s = sum(a.digits(base=p))
if factored:
return (s, out)
X = PolynomialRing(R, name='X').gen()
pi = R.ext(X**(p - 1) + p, names='pi').gen()
out *= pi**s
return out
def gauss_sum(a, p, f, prec=20, factored=False, algorithm='pari', parent=None):
r"""
Return the Gauss sum `g_q(a)` as a `p`-adic number.
The Gauss sum `g_q(a)` is defined by
.. MATH::
g_q(a)= \sum_{u\in F_q^*} \omega(u)^{-a} \zeta_q^u,
where `q = p^f`, `\omega` is the Teichmüller character and
`\zeta_q` is some arbitrary choice of primitive `q`-th root of
unity. The computation is adapted from the main theorem in Alain
Robert's paper *The Gross-Koblitz formula revisited*,
Rend. Sem. Mat. Univ. Padova 105 (2001), 157--170.
Let `p` be a prime, `f` a positive integer, `q=p^f`, and `\pi` be
the unique root of `f(x) = x^{p-1}+p` congruent to `\zeta_p - 1` modulo
`(\zeta_p - 1)^2`. Let `0\leq a < q-1`. Then the
Gross-Koblitz formula gives us the value of the Gauss sum `g_q(a)`
as a product of `p`-adic Gamma functions as follows:
.. MATH::
g_q(a) = -\pi^s \prod_{0\leq i < f} \Gamma_p(a^{(i)}/(q-1)),
where `s` is the sum of the digits of `a` in base `p` and the
`a^{(i)}` have `p`-adic expansions obtained from cyclic
permutations of that of `a`.
INPUT:
- ``a`` -- integer
- ``p`` -- prime
- ``f`` -- positive integer
- ``prec`` -- positive integer (optional, 20 by default)
- ``factored`` - boolean (optional, False by default)
- ``algorithm`` - flag passed to p-adic Gamma function (optional, "pari" by default)
OUTPUT:
If ``factored`` is ``False``, returns a `p`-adic number in an Eisenstein extension of `\QQ_p`.
This number has the form `pi^e * z` where `pi` is as above, `e` is some nonnegative
integer, and `z` is an element of `\ZZ_p`; if ``factored`` is ``True``, the pair `(e,z)`
is returned instead, and the Eisenstein extension is not formed.
.. NOTE::
This is based on GP code written by Adriana Salerno.
EXAMPLES:
In this example, we verify that `g_3(0) = -1`::
sage: from sage.rings.padics.misc import gauss_sum
sage: -gauss_sum(0,3,1)
1 + O(pi^40)
Next, we verify that `g_5(a) g_5(-a) = 5 (-1)^a`::
sage: from sage.rings.padics.misc import gauss_sum
sage: gauss_sum(2,5,1)^2-5
O(pi^84)
sage: gauss_sum(1,5,1)*gauss_sum(3,5,1)+5
O(pi^84)
Finally, we compute a non-trivial value::
sage: from sage.rings.padics.misc import gauss_sum
sage: gauss_sum(2,13,2)
6*pi^2 + 7*pi^14 + 11*pi^26 + 3*pi^62 + 6*pi^74 + 3*pi^86 + 5*pi^98 +
pi^110 + 7*pi^134 + 9*pi^146 + 4*pi^158 + 6*pi^170 + 4*pi^194 +
pi^206 + 6*pi^218 + 9*pi^230 + O(pi^242)
sage: gauss_sum(2,13,2,prec=5,factored=True)
(2, 6 + 6*13 + 10*13^2 + O(13^5))
.. SEEALSO::
- :func:`sage.arith.misc.gauss_sum` for general finite fields
- :meth:`sage.modular.dirichlet.DirichletCharacter.gauss_sum`
for prime finite fields
- :meth:`sage.modular.dirichlet.DirichletCharacter.gauss_sum_numerical`
for prime finite fields
"""
from sage.rings.padics.factory import Zp
from sage.rings.all import PolynomialRing
q = p**f
a = a % (q - 1)
if parent is None:
R = Zp(p, prec)
else:
R = parent
out = -R.one()
if a != 0:
t = R(1 / (q - 1))
for i in range(f):
out *= (a * t).gamma(algorithm)
a = (a * p) % (q - 1)
s = sum(a.digits(base=p))
if factored:
return s, out
X = PolynomialRing(R, name='X').gen()
pi = R.ext(X**(p - 1) + p, names='pi').gen()
out *= pi**s
return out
def gauss_sum(a, p, f, prec=20, factored=False, algorithm='pari', parent=None):
r"""
Return the Gauss sum `g_q(a)` as a `p`-adic number.
The Gauss sum `g_q(a)` is defined by
.. MATH::
g_q(a)= \sum_{u\in F_q^*} \omega(u)^{-a} \zeta_q^u,
where `q = p^f`, `\omega` is the Teichmüller character and
`\zeta_q` is some arbitrary choice of primitive `q`-th root of
unity. The computation is adapted from the main theorem in Alain
Robert's paper *The Gross-Koblitz formula revisited*,
Rend. Sem. Mat. Univ. Padova 105 (2001), 157--170.
Let `p` be a prime, `f` a positive integer, `q=p^f`, and `\pi` be
the unique root of `f(x) = x^{p-1}+p` congruent to `\zeta_p - 1` modulo
`(\zeta_p - 1)^2`. Let `0\leq a < q-1`. Then the
Gross-Koblitz formula gives us the value of the Gauss sum `g_q(a)`
as a product of `p`-adic Gamma functions as follows:
.. MATH::
g_q(a) = -\pi^s \prod_{0\leq i < f} \Gamma_p(a^{(i)}/(q-1)),
where `s` is the sum of the digits of `a` in base `p` and the
`a^{(i)}` have `p`-adic expansions obtained from cyclic
permutations of that of `a`.
INPUT:
- ``a`` -- integer
- ``p`` -- prime
- ``f`` -- positive integer
- ``prec`` -- positive integer (optional, 20 by default)
- ``factored`` - boolean (optional, False by default)
- ``algorithm`` - flag passed to p-adic Gamma function (optional, "pari" by default)
OUTPUT:
If ``factored`` is ``False``, returns a `p`-adic number in an Eisenstein extension of `\QQ_p`.
This number has the form `pi^e * z` where `pi` is as above, `e` is some nonnegative
integer, and `z` is an element of `\ZZ_p`; if ``factored`` is ``True``, the pair `(e,z)`
is returned instead, and the Eisenstein extension is not formed.
.. NOTE::
This is based on GP code written by Adriana Salerno.
EXAMPLES:
In this example, we verify that `g_3(0) = -1`::
sage: from sage.rings.padics.misc import gauss_sum
sage: -gauss_sum(0,3,1)
1 + O(pi^40)
Next, we verify that `g_5(a) g_5(-a) = 5 (-1)^a`::
sage: from sage.rings.padics.misc import gauss_sum
sage: gauss_sum(2,5,1)^2-5
O(pi^84)
sage: gauss_sum(1,5,1)*gauss_sum(3,5,1)+5
O(pi^84)
Finally, we compute a non-trivial value::
sage: from sage.rings.padics.misc import gauss_sum
sage: gauss_sum(2,13,2)
6*pi^2 + 7*pi^14 + 11*pi^26 + 3*pi^62 + 6*pi^74 + 3*pi^86 + 5*pi^98 +
pi^110 + 7*pi^134 + 9*pi^146 + 4*pi^158 + 6*pi^170 + 4*pi^194 +
pi^206 + 6*pi^218 + 9*pi^230 + O(pi^242)
sage: gauss_sum(2,13,2,prec=5,factored=True)
(2, 6 + 6*13 + 10*13^2 + O(13^5))
.. SEEALSO::
- :func:`sage.arith.misc.gauss_sum` for general finite fields
- :meth:`sage.modular.dirichlet.DirichletCharacter.gauss_sum`
for prime finite fields
- :meth:`sage.modular.dirichlet.DirichletCharacter.gauss_sum_numerical`
for prime finite fields
"""
from sage.rings.padics.factory import Zp
from sage.rings.all import PolynomialRing
q = p**f
a = a % (q-1)
if parent is None:
R = Zp(p, prec)
else:
R = parent
out = -R.one()
if a != 0:
t = R(1/(q-1))
for i in range(f):
out *= (a*t).gamma(algorithm)
a = (a*p) % (q-1)
s = sum(a.digits(base=p))
if factored:
return(s, out)
X = PolynomialRing(R, name='X').gen()
pi = R.ext(X**(p - 1) + p, names='pi').gen()
out *= pi**s
return out