Miscellaneous Functions

This file contains several miscellaneous functions used by \(p\)-adics.

  • gauss_sum – compute Gauss sums using the Gross-Koblitz formula.
  • min – a version of min that returns \(\infty\) on empty input.
  • max – a version of max that returns \(-\infty\) on empty input.

AUTHORS:

  • David Roe
  • Adriana Salerno
  • Ander Steele
  • Kiran Kedlaya (modified gauss_sum 2017/09)
sage.rings.padics.misc.gauss_sum(a, p, f, prec=20, factored=False, algorithm='pari', parent=None)

Return the Gauss sum \(g_q(a)\) as a \(p\)-adic number.

The Gauss sum \(g_q(a)\) is defined by

\[g_q(a)= \sum_{u\in F_q^*} \omega(u)^{-a} \zeta_q^u,\]

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:

\[g_q(a) = -\pi^s \prod_{0\leq i < f} \Gamma_p(a^{(i)}/(q-1)),\]

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))

See also

sage.rings.padics.misc.max(*L)

Return the maximum of the inputs, where the maximum of the empty list is \(-\infty\).

EXAMPLES:

sage: from sage.rings.padics.misc import max
sage: max()
-Infinity
sage: max(2,3)
3
sage.rings.padics.misc.min(*L)

Return the minimum of the inputs, where the minimum of the empty list is \(\infty\).

EXAMPLES:

sage: from sage.rings.padics.misc import min
sage: min()
+Infinity
sage: min(2,3)
2
sage.rings.padics.misc.precprint(prec_type, prec_cap, p)

String describing the precision mode on a p-adic ring or field.

EXAMPLES:

sage: from sage.rings.padics.misc import precprint
sage: precprint('capped-rel', 12, 2)
'with capped relative precision 12'
sage: precprint('capped-abs', 11, 3)
'with capped absolute precision 11'
sage: precprint('floating-point', 1234, 5)
'with floating precision 1234'
sage: precprint('fixed-mod', 1, 17)
'of fixed modulus 17^1'
sage.rings.padics.misc.trim_zeros(L)

Strips trailing zeros/empty lists from a list.

EXAMPLES:

sage: from sage.rings.padics.misc import trim_zeros
sage: trim_zeros([1,0,1,0])
[1, 0, 1]
sage: trim_zeros([[1],[],[2],[],[]])
[[1], [], [2]]
sage: trim_zeros([[],[]])
[]
sage: trim_zeros([])
[]

Zeros are also trimmed from nested lists (one deep):

sage: trim_zeros([[1,0]]) [[1]] sage: trim_zeros([[0],[1]]) [[], [1]]