This package provides a class for representing eta-products, which are meromorphic functions on modular curves of the form
where
is Dirichlet’s eta function
. These are useful
for obtaining explicit models of modular curves.
See trac ticket #3934 for background.
AUTHOR:
Return a list of CuspFamily objects corresponding to the cusps of
.
INPUT:
EXAMPLES:
sage: AllCusps(18)
[(Inf), (c_{2}), (c_{3,1}), (c_{3,2}), (c_{6,1}), (c_{6,2}), (c_{9}), (0)]
A family of elliptic curves parametrising a region of
.
Create the cusp of width d on X_0(N) corresponding to the family
of Tate curves
. Here
is a primitive root of unity of order
with
. The cusp doesn’t store zeta,
so we store an arbitrary label instead.
EXAMPLE:
sage: CuspFamily(8, 4)
(c_{4})
sage: CuspFamily(16, 4, '1')
(c_{4,1})
Return a string representation of self.
EXAMPLE:
sage: CuspFamily(16, 4, "1")._repr_()
'(c_{4,1})'
The level of this cusp.
EXAMPLES:
sage: e = CuspFamily(10, 1)
sage: e.level()
10
Return the corresponding element of
.
EXAMPLE:
sage: CuspFamily(10, 1).sage_cusp() # not implemented
Infinity
The width of this cusp.
EXAMPLES:
sage: e = CuspFamily(10, 1)
sage: e.width()
1
Create the group of eta products of the given level.
EXAMPLES:
sage: EtaGroup(12)
Group of eta products on X_0(12)
sage: EtaGroup(1/2)
...
TypeError: Level (=1/2) must be a positive integer
sage: EtaGroup(0)
...
ValueError: Level (=0) must be a positive integer
Compare self to other. Eta products compare first according to their levels, then according to their rdicts.
EXAMPLES:
sage: EtaProduct(2, {2:24,1:-24}) == 1
False
sage: EtaProduct(2, {2:24, 1:-24}) < EtaProduct(4, {2:24, 1:-24})
True
sage: EtaProduct(2, {2:24, 1:-24}) == EtaProduct(4, {2:24, 1:-24})
False
sage: EtaProduct(2, {2:24, 1:-24}) < EtaProduct(4, {2:48, 1:-48})
True
Create an eta product object. Usually called implicitly via EtaGroup_class.__call__ or the EtaProduct factory function.
EXAMPLE:
sage: EtaGroupElement(EtaGroup(8), {1:24, 2:-24})
Eta product of level 8 : (eta_1)^24 (eta_2)^-24
sage: g = _; g == loads(dumps(g))
True
Return .
EXAMPLES:
sage: eta1, eta2 = EtaGroup(4).basis()
sage: eta1 / eta2 # indirect doctest
Eta product of level 4 : (eta_1)^-16 (eta_2)^24 (eta_4)^-8
sage: (eta1 / eta2) * eta2 == eta1
True
Return the product of self and other.
EXAMPLES:
sage: eta1, eta2 = EtaGroup(4).basis() # indirect doctest
sage: eta1 * eta2
Eta product of level 4 : (eta_2)^24 (eta_4)^-24
Return the string representation of self.
EXAMPLES:
sage: EtaProduct(3, {3:12, 1:-12})._repr_()
'Eta product of level 3 : (eta_1)^-12 (eta_3)^12'
A short string representation of self, which doesn’t specify the level.
EXAMPLES:
sage: EtaProduct(3, {3:12, 1:-12})._short_repr()
'(eta_1)^-12 (eta_3)^12'
Return the degree of self as a map
, which is equal to the sum of
all the positive coefficients in the divisor of self.
EXAMPLES:
sage: e = EtaProduct(12, {1:-336, 2:576, 3:696, 4:-216, 6:-576, 12:-144})
sage: e.degree()
230
Return the divisor of self, as a formal sum of CuspFamily objects.
EXAMPLES:
sage: e = EtaProduct(12, {1:-336, 2:576, 3:696, 4:-216, 6:-576, 12:-144})
sage: e.divisor() # FormalSum seems to print things in a random order?
-131*(Inf) - 50*(c_{2}) + 11*(0) + 50*(c_{6}) + 169*(c_{4}) - 49*(c_{3})
sage: e = EtaProduct(2^8, {8:1,32:-1})
sage: e.divisor() # random
-(c_{2}) - (Inf) - (c_{8,2}) - (c_{8,3}) - (c_{8,4}) - (c_{4,2}) - (c_{8,1}) - (c_{4,1}) + (c_{32,4}) + (c_{32,3}) + (c_{64,1}) + (0) + (c_{32,2}) + (c_{64,2}) + (c_{128}) + (c_{32,1})
Return the level of this eta product.
EXAMPLES:
sage: e = EtaProduct(3, {3:12, 1:-12})
sage: e.level()
3
sage: EtaProduct(12, {6:6, 2:-6}).level() # not the lcm of the d's
12
sage: EtaProduct(36, {6:6, 2:-6}).level() # not minimal
36
Return the order of vanishing of self at the given cusp.
INPUT:
OUTPUT:
EXAMPLES:
sage: e = EtaProduct(2, {2:24, 1:-24})
sage: e.order_at_cusp(CuspFamily(2, 1)) # cusp at infinity
1
sage: e.order_at_cusp(CuspFamily(2, 2)) # cusp 0
-1
The q-expansion of self at the cusp at infinity.
INPUT:
OUTPUT:
ALGORITHM: Calculates eta to (n/m) terms, where m is the smallest integer dividing self.level() such that self.r(m) != 0. Then multiplies.
EXAMPLES:
sage: EtaProduct(36, {6:6, 2:-6}).q_expansion(10)
q + 6*q^3 + 27*q^5 + 92*q^7 + 279*q^9 + O(q^11)
sage: R.<q> = ZZ[[]]
sage: EtaProduct(2,{2:24,1:-24}).q_expansion(100) == delta_qexp(101)(q^2)/delta_qexp(101)(q)
True
Alias for self.q_expansion().
EXAMPLES:
sage: e = EtaProduct(36, {6:8, 3:-8})
sage: e.qexp(10)
q + 8*q^4 + 36*q^7 + O(q^10)
sage: e.qexp(30) == e.q_expansion(30)
True
Return the exponent of
in self.
EXAMPLES:
sage: e = EtaProduct(12, {2:24, 3:-24})
sage: e.r(3)
-24
sage: e.r(4)
0
The group of eta products of a given level under multiplication.
Create an element of this group (an eta product object) with exponents from the given dictionary. See the docstring for the EtaProduct() factory function for how dict is used.
EXAMPLE:
sage: EtaGroup(2).__call__({1:24, 2:-24})
Eta product of level 2 : (eta_1)^24 (eta_2)^-24
Compare self to other. If other is not an EtaGroup, compare by type; otherwise compare by level. EtaGroups of the same level compare as identical.
EXAMPLE:
sage: EtaGroup(12) == 12
False
sage: EtaGroup(12) < EtaGroup(13)
True
sage: EtaGroup(12) == EtaGroup(12)
True
Create the group of eta products of a given level, which must be a positive integer.
Return the data used to construct self. Used for pickling.
EXAMPLE:
sage: EtaGroup(13).__reduce__()
(<function EtaGroup at ...>, (13,))
String representation of self.
EXAMPLE:
sage: EtaGroup(12)._repr_()
'Group of eta products on X_0(12)'
Produce a basis for the free abelian group of eta-products of level N (under multiplication), attempting to find basis vectors of the smallest possible degree.
INPUT:
EXAMPLE:
sage: EtaGroup(5).basis()
[Eta product of level 5 : (eta_1)^6 (eta_5)^-6]
sage: EtaGroup(12).basis()
[Eta product of level 12 : (eta_1)^2 (eta_2)^1 (eta_3)^2 (eta_4)^-1 (eta_6)^-7 (eta_12)^3,
Eta product of level 12 : (eta_1)^-2 (eta_2)^3 (eta_3)^6 (eta_4)^-1 (eta_6)^-9 (eta_12)^3,
Eta product of level 12 : (eta_1)^-3 (eta_2)^2 (eta_3)^1 (eta_4)^-1 (eta_6)^-2 (eta_12)^3,
Eta product of level 12 : (eta_1)^1 (eta_2)^-1 (eta_3)^-3 (eta_4)^-2 (eta_6)^7 (eta_12)^-2,
Eta product of level 12 : (eta_1)^-6 (eta_2)^9 (eta_3)^2 (eta_4)^-3 (eta_6)^-3 (eta_12)^1]
sage: EtaGroup(12).basis(reduce=False) # much bigger coefficients
[Eta product of level 12 : (eta_2)^24 (eta_12)^-24,
Eta product of level 12 : (eta_1)^-336 (eta_2)^576 (eta_3)^696 (eta_4)^-216 (eta_6)^-576 (eta_12)^-144,
Eta product of level 12 : (eta_1)^-8 (eta_2)^-2 (eta_6)^2 (eta_12)^8,
Eta product of level 12 : (eta_1)^1 (eta_2)^9 (eta_3)^13 (eta_4)^-4 (eta_6)^-15 (eta_12)^-4,
Eta product of level 12 : (eta_1)^15 (eta_2)^-24 (eta_3)^-29 (eta_4)^9 (eta_6)^24 (eta_12)^5]
ALGORITHM: An eta product of level is uniquely
determined by the integers
for
with
, since
. The valid
are those that satisfy two congruences modulo 24,
and one congruence modulo 2 for every prime divisor of N. We beef
up the congruences modulo 2 to congruences modulo 24 by multiplying
by 12. To calculate the kernel of the ensuing map
we lift it arbitrarily to an integer matrix and calculate its Smith
normal form. This gives a basis for the lattice.
This lattice typically contains “large” elements, so by default we pass it to the reduce_basis() function which performs LLL-reduction to give a more manageable basis.
Return the level of self. EXAMPLES:
sage: EtaGroup(10).level()
10
Produce a more manageable basis via LLL-reduction.
INPUT:
OUTPUT:
ALGORITHM: We define the norm of an eta-product to be the
norm of its divisor (as an element of the free
-module with the cusps as basis and the
standard inner product). Applying LLL-reduction to this gives a
basis of hopefully more tractable elements. Of course we’d like to
use the
norm as this is just twice the degree, which
is a much more natural invariant, but
norm is easier
to work with!
EXAMPLES:
sage: EtaGroup(4).reduce_basis([ EtaProduct(4, {1:8,2:24,4:-32}), EtaProduct(4, {1:8, 4:-8})])
[Eta product of level 4 : (eta_1)^8 (eta_4)^-8,
Eta product of level 4 : (eta_1)^-8 (eta_2)^24 (eta_4)^-16]
Create an EtaGroupElement object representing the function
. Checks the criteria
of Ligozat to ensure that this product really is the q-expansion of
a meromorphic function on X_0(N).
INPUT:
OUTPUT:
Note
The dictionary dict does not uniquely specify N. It is
possible for two EtaGroupElements with different ‘s to
be created with the same dictionary, and these represent different
objects (although they will have the same
-expansion at
the cusp
).
EXAMPLES:
sage: EtaProduct(3, {3:12, 1:-12})
Eta product of level 3 : (eta_1)^-12 (eta_3)^12
sage: EtaProduct(3, {3:6, 1:-6})
...
ValueError: sum d r_d (=12) is not 0 mod 24
sage: EtaProduct(3, {4:6, 1:-6})
...
ValueError: 4 does not divide 3
Helper function used by eta_poly_relations. Finds a basis for the
space of linear relations between the first qexp_terms of the
-expansions of the monomials
for
,
and calculates a Groebner basis for the ideal generated by these
relations.
Liable to return meaningless results if qexp_terms isn’t at least
where
as then 1 will be in the ideal.
EXAMPLE:
sage: from sage.modular.etaproducts import _eta_relations_helper
sage: r,s = EtaGroup(4).basis()
sage: _eta_relations_helper(r,s,4,100,['a','b'],False)
[a - b - 16]
sage: _eta_relations_helper(EtaProduct(26, {2:2,13:2,26:-2,1:-2}),EtaProduct(26, {2:4,13:2,26:-4,1:-2}),3,12,['a','b'],False) # not enough terms, will return rubbish
[1]
Find polynomial relations between eta products.
INPUTS:
OUTPUTS: a list of polynomials which is a Groebner basis for the part of the ideal of relations between eta_elements which is generated by elements up to the given degree; or None, if no relations were found.
ALGORITHM: An expression of the form
is zero if and
only if it vanishes at the cusp infinity to degree at least
. For all terms up to
in the
-expansion of this expression to be zero is a
system of
linear equations in
coefficients, where
is the number of nonzero negative
coefficients that can appear.
Solving these equations and calculating a basis for the solution space gives us a set of polynomial relations, but this is generally far from a minimal generating set for the ideal, so we calculate a Groebner basis.
As a test, we calculate five extra terms of -expansion
and check that this doesn’t change the answer.
EXAMPLES:
sage: t = EtaProduct(26, {2:2,13:2,26:-2,1:-2})
sage: u = EtaProduct(26, {2:4,13:2,26:-4,1:-2})
sage: eta_poly_relations([t, u], 3)
sage: eta_poly_relations([t, u], 4)
[x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]
Use verbose=True to see the details of the computation:
sage: eta_poly_relations([t, u], 3, verbose=True)
Trying to find a relation of degree 3
Lowest order of a term at infinity = -12
Highest possible degree of a term = 15
Trying all coefficients from q^-12 to q^15 inclusive
No polynomial relation of order 3 valid for 28 terms
Check: Trying all coefficients from q^-12 to q^20 inclusive
No polynomial relation of order 3 valid for 33 terms
sage: eta_poly_relations([t, u], 4, verbose=True)
Trying to find a relation of degree 4
Lowest order of a term at infinity = -16
Highest possible degree of a term = 20
Trying all coefficients from q^-16 to q^20 inclusive
Check: Trying all coefficients from q^-16 to q^25 inclusive
[x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]
Return the number of cusps on of width d.
INPUT:
EXAMPLES:
sage: [num_cusps_of_width(18,d) for d in divisors(18)]
[1, 1, 2, 2, 1, 1]
Return the q-expansion of , where
is Dedekind’s function
as an element of ps_ring, to precision n. Completely naive algorithm.
INPUT:
OUTPUT: An element of ps_ring which is the q-expansion of
truncated to n terms.
ALGORITHM: Multiply out the product
. Could perhaps be sped-up by
using the identity
but I’m lazy.
EXAMPLES:
sage: qexp_eta(ZZ[['q']], 100)
1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + q^22 + q^26 - q^35 - q^40 + q^51 + q^57 - q^70 - q^77 + q^92 + O(q^100)