def statement_simple(desc, g, str_field='K'):
_index_dict_ = index_dictionary()
factors_QQ = desc[_index_dict_['factors_QQ']]
if len(factors_QQ) == 1:
factor_QQ = factors_QQ[0]
m = factor_QQ[_index_dict_['m']]
disc = factor_QQ[_index_dict_['disc']]
if m == 1 or disc != 1:
return "simple"
return "not simple"
def statement_gl2(desc, g, str_field='K'):
_index_dict_ = index_dictionary()
factors_QQ = desc[_index_dict_['factors_QQ']]
dimsum = 0
for factor_QQ in factors_QQ:
dim_base = len(factor_QQ[_index_dict_['center']]) - 1
dim_sqrt = factor_QQ[_index_dict_['d']] * factor_QQ[_index_dict_['m']]
dimsum += dim_base * dim_sqrt
if dimsum == g:
return "of GL_2-type"
return "not of GL_2-type"
def statement_factors_ZZ_index(factors_QQ, desc_ZZ, str_field='K'):
_index_dict_ = index_dictionary()
# TODO: Assumes g <= 3
index = desc_ZZ[_index_dict_['index']]
if len(factors_QQ) == 1:
factor_QQ = factors_QQ[0]
albert_type = factor_QQ[_index_dict_['albert_type']]
m = factor_QQ[_index_dict_['m']]
if albert_type == 'I' and m == 1:
desc_field = factor_QQ[_index_dict_['center']]
return pretty_print_ring(desc_field, index)
return "Sub (End (J_%s) ox QQ, %s)" % (str_field, index)
def statement_eichler(desc, g, str_field='K'):
_index_dict_ = index_dictionary()
factors_QQ = desc[_index_dict_['factors_QQ']]
desc_ZZ = desc[_index_dict_['desc_ZZ']]
if len(factors_QQ) == 1:
factor_QQ = factors_QQ[0]
albert_type = factor_QQ[_index_dict_['albert_type']]
is_eichler = desc_ZZ[_index_dict_['is_eichler']]
if is_eichler == 1:
return "(Eichler)"
elif is_eichler == 0:
return "(non-Eichler)"
return ""
def statement_cm(desc, g, str_field='K'):
_index_dict_ = index_dictionary()
factors_QQ = desc[_index_dict_['factors_QQ']]
dimsum = 0
for factor_QQ in factors_QQ:
albert_type = factor_QQ[_index_dict_['albert_type']]
dim_base = len(factor_QQ[_index_dict_['center']]) - 1
m = factor_QQ[_index_dict_['m']]
if albert_type == 'IV':
dimsum += dim_base * m
if (dimsum // 2) == g:
return "(CM)"
return ""
def pretty_print_lattice_description(lat, g):
_index_dict_ = index_dictionary()
statements = []
base = lat[_index_dict_['base'] - 1]
entries = lat[_index_dict_['entries'] - 1]
for entry in entries:
statement = ''
field = entry[_index_dict_['field'] - 1]
statement += "Over subfield %s:\n" % pretty_print_field(field, base)
structure = entry[_index_dict_['structure'] - 1]
statement += statements_all(structure, g)
statements.append(statement)
return '\n\n'.join(statements)
def statement_endomorphisms_ZZ(desc, g, str_field='K'):
_index_dict_ = index_dictionary()
factors_QQ = desc[_index_dict_['factors_QQ']]
desc_ZZ = desc[_index_dict_['desc_ZZ']]
if desc_ZZ[_index_dict_['index']] == 1:
statements = [
statement_factor_ZZ_maximal(factor_QQ, desc_ZZ)
for factor_QQ in factors_QQ
]
statement = ' x '.join(statements)
else:
statement = statement_factors_ZZ_index(factors_QQ, desc_ZZ)
return "End (J_%s): " % str_field + statement
def statement_factor_ZZ_maximal(factor_QQ, desc_ZZ, str_field='K'):
_index_dict_ = index_dictionary()
# TODO: Assumes g <= 3
albert_type = factor_QQ[_index_dict_['albert_type']]
field_pretty = pretty_print_field(factor_QQ[_index_dict_['center']],
[-1, 1])
ring_pretty = pretty_print_ring(factor_QQ[_index_dict_['center']], 1)
disc = factor_QQ[_index_dict_['disc']]
m = factor_QQ[_index_dict_['m']]
if albert_type == 'I' or albert_type == 'IV':
if m == 1:
return ring_pretty
# TODO: Next line can be done in greater generality over PIDs, but to
# found out whether we are in such a situation costs time, so omitted
# for now.
elif (disc == 1) and field_pretty == 'QQ':
return 'M_%s (%s)' % (m, ring_pretty)
return 'Max (%s)' % statement_factor_QQ(factor_QQ)
def statement_factor_QQ(factor_QQ):
_index_dict_ = index_dictionary()
# TODO: Assumes g <= 3, in general we need powers
albert_type = factor_QQ[_index_dict_['albert_type']]
d = factor_QQ[_index_dict_['d']]
disc = factor_QQ[_index_dict_['disc']]
m = factor_QQ[_index_dict_['m']]
str_center = pretty_print_field(factor_QQ[_index_dict_['center']], [-1, 1])
str_d = str(d)
str_disc = str(disc)
str_m = str(m)
if albert_type == 'I':
if m == 1:
statement = str_center
else:
statement = "M_%s (%s)" % (str_m, str_center)
elif albert_type == 'II':
if d == 2:
statement = "IndefQuat (%s, %s)" % (str_center, str_disc)
else:
statement = "IndefAlg_%s (%s, %s)" % (str_d, str_center, str_disc)
elif albert_type == 'III':
if d == 2:
statement = "DefQuat (%s, %s)" % (str_center, str_disc)
else:
statement = "DefAlg_%s (%s, %s)" % (str_d, str_center, str_disc)
elif albert_type == 'IV':
if m == 1:
statement = str_center
elif d == 1 and m == 2:
statement = "M_%s (%s)" % (str_m, str_center)
elif m == 1 and d == 2:
statement = "Quat (%s, %s)" % (str_center, str_disc)
else:
statement = "Alg_%s (%s, %s)" % (str_d, str_center, str_disc)
return statement
def statement_factor_QQ(factor_QQ):
_index_dict_ = index_dictionary()
# TODO: Assumes g <= 3
dim_sqrt = factor_QQ[_index_dict_['dim_sqrt']]
disc = factor_QQ[_index_dict_['disc']]
albert_type = factor_QQ[_index_dict_['albert_type']]
str_center = pretty_print_field(factor_QQ[_index_dict_['center']], [-1, 1])
str_dim_sqrt = str(dim_sqrt)
str_disc = str(disc)
if albert_type == 'I':
if dim_sqrt == 1:
statement = str_center
else:
statement = "M_%s (%s)" % (str_dim_sqrt, str_center)
elif albert_type == 'II':
if dim_sqrt == 2:
statement = "IndefQuat (%s, %s)" % (str_center, str_disc)
else:
statement = "IndefAlg_%s (%s, %s)" % (str_dim_sqrt, str_center,
str_disc)
elif albert_type == 'III':
if dim_sqrt == 2:
statement = "DefQuat (%s, %s)" % (str_center, str_disc)
else:
statement = "DefAlg_%s (%s, %s)" % (str_dim_sqrt, str_center,
str_disc)
elif albert_type == 'IV':
if dim_sqrt == 1:
statement = str_center
elif disc == 1:
statement = "M_%s (%s)" % (str_dim_sqrt, str_center)
elif dim_sqrt == 2:
statement = "Quat (%s, %s)" % (str_center, str_disc)
else:
statement = "Alg_%s (%s, %s)" % (str_dim_sqrt, str_center,
str_disc)
return statement
def statement_endomorphisms_QQ(desc, g, str_field='K'):
_index_dict_ = index_dictionary()
factors_QQ = desc[_index_dict_['factors_QQ']]
statements = [statement_factor_QQ(factor_QQ) for factor_QQ in factors_QQ]
statement = ' x '.join(statements)
return "End (J_%s) ox QQ: " % str_field + statement
def statement_sato_tate_group(desc, g, str_field='K'):
_index_dict_ = index_dictionary()
sato_tate = desc[_index_dict_['sato_tate']]
if sato_tate == "" or sato_tate == "undef":
sato_tate = "not classified yet"
return "Sato-Tate group: %s" % sato_tate
def statement_endomorphisms_RR(desc, g, str_field='K'):
_index_dict_ = index_dictionary()
return "End (J_%s) ox RR: %s" % (str_field, ' x '.join(
desc[_index_dict_['desc_RR']]))
