def get_pgl2q_cuspidal_characters(q):
field = FieldFromInteger.from_q(q)
square_field = SquareExtensionField.from_base_field(field)
n2 = q + 1
square_field_characters = [FieldCharacter(square_field, i, n2) for i in range(1, n2 // 2)]
cuspidal_characters = [PGL2CuspidalCharacter(q, field_character) for field_character in square_field_characters]
return cuspidal_characters
def get_psl2q_characters(q) -> List[Character]:
field = FieldFromInteger.from_q(q)
standard_character = PSL2StandardCharacter(q)
characters = list()
characters.append(TrivialCharacter())
characters.append(standard_character)
n = q-1
field_characters = [FieldCharacter(field, i, n) for i in range(2, n//2, 2)]
normal_diagonalizable_characters = [PSL2DiagonalizableCharacter(q, field_character)
for field_character in field_characters]
characters.extend(normal_diagonalizable_characters)
square_field = SquareExtensionField.from_base_field(field)
n2 = q+1
square_field_characters = [FieldCharacter(square_field, i, n2) for i in range(1, (n2//2+1)//2)]
cuspidal_characters = [PSL2CuspidalCharacter(q, field_character)
for field_character in square_field_characters]
characters.extend(cuspidal_characters)
if q % 4 == 1:
oscillator_class = PSL2EvenOscillatorCharacter
else:
oscillator_class = PSL2OddOscillatorCharacter
characters.append(oscillator_class(q, 1))
characters.append(oscillator_class(q, -1))
return characters
def get_pgl2q_characters(q) -> List[Character]:
field = FieldFromInteger.from_q(q)
sign_character = PGL2SignCharacter()
standard_character = PGL2StandardCharacter(q)
characters = []
characters.extend([
TrivialCharacter(),
sign_character,
standard_character,
ProductCharacter(standard_character, sign_character),
])
n = q-1
field_characters = [FieldCharacter(field, i, n) for i in range(1, n//2)]
normal_diagonalizable_characters = [PGL2DiagonalizableCharacter(q, field_character)
for field_character in field_characters]
characters.extend(normal_diagonalizable_characters)
cuspidal_characters = get_pgl2q_cuspidal_characters(q)
characters.extend(cuspidal_characters)
return characters
def pxl2_lubotzky_prime_based_subset(q, p):
assert (q % 4 == 1)
assert (p % 4 == 1)
assert (p != q)
field = FieldFromInteger.from_q(q)
pgl2 = PGL2(q)
max_p_sqrt = int(p**0.5) + 1
a_s = [(a, b, c, d) for a, (b, c, d) in product(
range(1, max_p_sqrt, 2), product(range(0, max_p_sqrt, 2), repeat=3))
if a * a + b * b + c * c + d * d == p]
a_s += [(a, -b, c, d) for a, b, c, d in a_s]
a_s += [(a, b, -c, d) for a, b, c, d in a_s]
a_s += [(a, b, c, -d) for a, b, c, d in a_s]
a_s = set(a_s)
a_s = [(field.create_element(a), field.create_element(b),
field.create_element(c), field.create_element(d))
for a, b, c, d in a_s]
i = (-field.one()).sqrt()
matrices = [
pgl2.create([[a + i * b, c + i * d], [-c + i * d, a - i * b]])
for a, b, c, d in a_s
]
return matrices
def __init__(self, q):
self._base_field = FieldFromInteger.from_q(q)
self._q = q
self._square_extension = SquareExtensionField.from_base_field(
self._base_field)
def __init__(self, n, q):
self._field = FieldFromInteger.from_q(q)
self.n = n