Eta-products on modular curves \(X_0(N)\)

This package provides a class for representing eta-products, which are meromorphic functions on modular curves of the form

\[\prod_{d | N} \eta(q^d)^{r_d}\]

where \(\eta(q)\) is Dirichlet’s eta function \(q^{1/24} \prod_{n = 1}^\infty(1-q^n)\). These are useful for obtaining explicit models of modular curves.

See trac ticket #3934 for background.

AUTHOR:

• David Loeffler (2008-08-22): initial version

sage.modular.etaproducts.AllCusps(N)

Return a list of CuspFamily objects corresponding to the cusps of \(X_0(N)\).

INPUT:

• N - (integer): the level

EXAMPLES:

sage: AllCusps(18)
[(Inf), (c_{2}), (c_{3,1}), (c_{3,2}), (c_{6,1}), (c_{6,2}), (c_{9}), (0)]
sage: AllCusps(0)
Traceback (most recent call last):
...
ValueError: N must be positive

class sage.modular.etaproducts.CuspFamily(N, width, label=None)

Bases: sage.structure.sage_object.SageObject

A family of elliptic curves parametrising a region of \(X_0(N)\).

level()

The level of this cusp.

EXAMPLES:

sage: e = CuspFamily(10, 1)
sage: e.level()
10

sage_cusp()

Return the corresponding element of \(\mathbb{P}^1(\QQ)\).

EXAMPLES:

sage: CuspFamily(10, 1).sage_cusp() # not implemented
Infinity

width()

The width of this cusp.

EXAMPLES:

sage: e = CuspFamily(10, 1)
sage: e.width()
1

sage.modular.etaproducts.EtaGroup(level)

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)
Traceback (most recent call last):
...
TypeError: Level (=1/2) must be a positive integer
sage: EtaGroup(0)
Traceback (most recent call last):
...
ValueError: Level (=0) must be a positive integer

class sage.modular.etaproducts.EtaGroupElement(parent, rdict)

Bases: sage.structure.element.MultiplicativeGroupElement

Create an eta product object. Usually called implicitly via EtaGroup_class.__call__ or the EtaProduct factory function.

EXAMPLES:

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

degree()

Return the degree of self as a map \(X_0(N) \to \mathbb{P}^1\), 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

divisor()

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})

level()

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

order_at_cusp(cusp)

Return the order of vanishing of self at the given cusp.

INPUT:

• cusp – a CuspFamily object

OUTPUT:

• an integer

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

q_expansion(n)

The q-expansion of self at the cusp at infinity.

INPUT:

• n (integer): number of terms to calculate

OUTPUT:

• a power series over \(\ZZ\) in the variable \(q\), with a relative precision of \(1 + O(q^n)\).

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

qexp(n)

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

r(d)

Return the exponent \(r_d\) of \(\eta(q^d)\) in self.

EXAMPLES:

sage: e = EtaProduct(12, {2:24, 3:-24})
sage: e.r(3)
-24
sage: e.r(4)
0

class sage.modular.etaproducts.EtaGroup_class(level)

Bases: sage.groups.old.AbelianGroup

The group of eta products of a given level under multiplication.

basis(reduce=True)

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:

• reduce - a boolean (default True) indicating whether or not to apply LLL-reduction to the calculated basis

EXAMPLES:

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)^-4 (eta_2)^2 (eta_3)^4 (eta_6)^-2,
Eta product of level 12 : (eta_1)^-1 (eta_2)^3 (eta_3)^3 (eta_4)^-2 (eta_6)^-9 (eta_12)^6,
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 \(N\) is uniquely determined by the integers \(r_d\) for \(d | N\) with \(d < N\), since \(\sum_{d | N} r_d = 0\). The valid \(r_d\) 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 \(\ZZ^m \to (\ZZ/24\ZZ)^n\) 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.

level()

Return the level of self. EXAMPLES:

sage: EtaGroup(10).level()
10

one()

Return the identity element of self.

EXAMPLES:

sage: EtaGroup(12).one()
Eta product of level 12 : 1

reduce_basis(long_etas)

Produce a more manageable basis via LLL-reduction.

INPUT:

• long_etas - a list of EtaGroupElement objects (which should all be of the same level)

OUTPUT:

• a new list of EtaGroupElement objects having hopefully smaller norm

ALGORITHM: We define the norm of an eta-product to be the \(L^2\) norm of its divisor (as an element of the free \(\ZZ\)-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 \(L^1\) norm as this is just twice the degree, which is a much more natural invariant, but \(L^2\) 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]

sage.modular.etaproducts.EtaProduct(level, dict)

Create an EtaGroupElement object representing the function \(\prod_{d | N} \eta(q^d)^{r_d}\). Checks the criteria of Ligozat to ensure that this product really is the q-expansion of a meromorphic function on X_0(N).

INPUT:

• level - (integer): the N such that this eta product is a function on X_0(N).
• dict - (dictionary): a dictionary indexed by divisors of N such that the coefficient of \(\eta(q^d)\) is r[d]. Only nonzero coefficients need be specified. If Ligozat’s criteria are not satisfied, a ValueError will be raised.

OUTPUT:

• an EtaGroupElement object, whose parent is the EtaGroup of level N and whose coefficients are the given dictionary.

Note

The dictionary dict does not uniquely specify N. It is possible for two EtaGroupElements with different \(N\)’s to be created with the same dictionary, and these represent different objects (although they will have the same \(q\)-expansion at the cusp \(\infty\)).

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})
Traceback (most recent call last):
...
ValueError: sum d r_d (=12) is not 0 mod 24
sage: EtaProduct(3, {4:6, 1:-6})
Traceback (most recent call last):
...
ValueError: 4 does not divide 3

sage.modular.etaproducts.eta_poly_relations(eta_elements, degree, labels=['x1', 'x2'], verbose=False)

Find polynomial relations between eta products.

INPUT:

• eta_elements - (list): a list of EtaGroupElement objects. Not implemented unless this list has precisely two elements. degree
• degree - (integer): the maximal degree of polynomial to look for.
• labels - (list of strings): labels to use for the polynomial returned.
• verbose` - (boolean, default False): if True, prints information as it goes.

OUTPUT: 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 \(\sum_{0 \le i,j \le d} a_{ij} x^i y^j\) is zero if and only if it vanishes at the cusp infinity to degree at least \(v = d(deg(x) + deg(y))\). For all terms up to \(q^v\) in the \(q\)-expansion of this expression to be zero is a system of \(v + k\) linear equations in \(d^2\) coefficients, where \(k\) 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 \(q\)-expansion and check that this doesn’t change the answer.

EXAMPLES:

sage: from sage.modular.etaproducts import eta_poly_relations
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]

sage.modular.etaproducts.num_cusps_of_width(N, d)

Return the number of cusps on \(X_0(N)\) of width d.

INPUT:

• N - (integer): the level
• d - (integer): an integer dividing N, the cusp width

EXAMPLES:

sage: from sage.modular.etaproducts import num_cusps_of_width
sage: [num_cusps_of_width(18,d) for d in divisors(18)]
[1, 1, 2, 2, 1, 1]
sage: num_cusps_of_width(4,8)
Traceback (most recent call last):
...
ValueError: N and d must be positive integers with d|N

sage.modular.etaproducts.qexp_eta(ps_ring, prec)

Return the q-expansion of \(\eta(q) / q^{1/24}\), where \(\eta(q)\) is Dedekind’s function

\[\eta(q) = q^{1/24}\prod_{n=1}^\infty (1-q^n),\]

as an element of ps_ring, to precision prec.

INPUT:

• ps_ring – (PowerSeriesRing): a power series ring
• prec – (integer): the number of terms to compute

OUTPUT: An element of ps_ring which is the q-expansion of \(\eta(q)/q^{1/24}\) truncated to prec terms.

ALGORITHM: We use the Euler identity

\[\eta(q) = q^{1/24}( 1 + \sum_{n \ge 1} (-1)^n (q^{n(3n+1)/2} + q^{n(3n-1)/2})\]

to compute the expansion.

EXAMPLES:

sage: from sage.modular.etaproducts import qexp_eta
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)