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Implement hypergeometric Euler factors at t=1 #38322

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merged 3 commits into from
Jul 24, 2024
Merged

Implement hypergeometric Euler factors at t=1 #38322

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bf8a356
Implement hypergeometric traces, Euler factors at t=1
kedlaya Jul 1, 2024
cc8671e
Fix whitespace issue
kedlaya Jul 1, 2024
eb236ab
Incidental docstring formatting fixes (add class links)
kedlaya Jul 6, 2024
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21 changes: 16 additions & 5 deletions src/sage/modular/hypergeometric_motive.py
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Original file line number Diff line number Diff line change
Expand Up @@ -1496,9 +1496,11 @@ def H_value(self, p, f, t, ring=None):
Using instead :class:`UniversalCyclotomicfield`, this is much
slower than the `p`-adic version :meth:`padic_H_value`.

Unlike in :meth:`padic_H_value`, tame and wild primes are not supported.

EXAMPLES:

With values in the UniversalCyclotomicField (slow)::
With values in the ``UniversalCyclotomicField`` (slow)::
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sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(alpha_beta=([1/2]*4, [0]*4))
Expand All @@ -1513,7 +1515,7 @@ def H_value(self, p, f, t, ring=None):
sage: [H.H_value(13,i,-1) for i in range(1,3)] # not tested
[-84, -1420]

With values in ComplexField::
With values in ``ComplexField``::
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sage: [H.H_value(5,i,-1, ComplexField(60)) for i in range(1,3)]
[-4, 276]
Expand Down Expand Up @@ -1750,7 +1752,7 @@ def euler_factor(self, t, p, deg=None, cache_p=False):

INPUT:

- ``t`` -- rational number, not 0 or 1
- ``t`` -- rational number, not 0

- ``p`` -- prime number of good reduction

Expand Down Expand Up @@ -1837,6 +1839,15 @@ def euler_factor(self, t, p, deg=None, cache_p=False):
sage: H.euler_factor(1/7^4, 7)
7*T^3 + 7*T^2 + T + 1

This is an example with `t = 1`::

sage: H = Hyp(cyclotomic=[[4,2], [3,1]])
sage: H.euler_factor(1, 7)
-T^2 + 1
sage: H = Hyp(cyclotomic=[[5], [1,1,1,1]])
sage: H.euler_factor(1, 7)
343*T^2 - 6*T + 1

TESTS::

sage: H1 = Hyp(alpha_beta=([1,1,1], [1/2,1/2,1/2]))
Expand Down Expand Up @@ -1926,7 +1937,7 @@ def euler_factor(self, t, p, deg=None, cache_p=False):
- [Watkins]_
"""
t = QQ(t)
if t in [0, 1]:
if t == 0:
raise ValueError('invalid t')
if not is_prime(p):
raise ValueError('p not prime')
Expand Down Expand Up @@ -2003,7 +2014,7 @@ def euler_factor(self, t, p, deg=None, cache_p=False):
ans = characteristic_polynomial_from_traces(traces, d, p, w, sign, deg=deg)

# In the multiplicative case, we sometimes need to add extra factors.
if typ == "mult":
if typ == "mult" and t != 1:
if self.degree() % 2 == 0:
ans *= tmp
if w % 2 == 0 and (t-1).valuation(p) % 2 == 0:
Expand Down
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