Expansion at infinity for rank 2#
In analogy with the classical theory, any Drinfeld modular form \(f:\Omega^2(\mathbb{C}_{\infty}) \rightarrow \mathbb{C}_{\infty}\) admits an expansion at infinity of the form:
where \(u(w) := 1/e(w)\) and \(e(w)\) is the exponential of the Carlitz module \(T\mapsto T + \tau\).
We say that a Drinfeld modular forms of weight \(k\) admits an \(A\)-expansion if there exists an integer \(n\) and coefficients \(c_{a}(f)\) such that
where \(G_n(X)\) is the \(n\)-th Goss polynomial of the Carlitz module.
Petrov showed that there exists an infinite family of Drinfeld modular forms with \(A\)-expansion:
provided that \(k\) and \(n\) are integers such that \(k - 2n \equiv 0\) modulo \(q - 1\) and \(n \leq p^{v_p(k - n)}\). The set of all forms \(f_{k, n}\) will be called the Petrov family.
This module defines functions that compute the expansion at infinity of any \(A\)-expansion.
EXAMPLES:
sage: from drinfeld_modular_forms import compute_petrov_expansion
sage: q = 3
sage: D = compute_petrov_expansion(q + 1, 1, q); D
u + u^5 + ((2*T^3+T)*u^7) + O(u^8)
It is possible to compute on demands any coefficients of the above series:
sage: D[59] # 59-th coefficient
2*T^27 + T^9 + T^3 + 2*T
sage: D[0:10] # first 10 coefficients
[0, 1, 0, 0, 0, 1, 0, 2*T^3 + T, 0, 1]
AUTHORS:
David Ayotte (2021): initial version
- drinfeld_modular_forms.expansions.coefficient_A_expansion(coeff_stream, n, i, param)#
Return the \(i\)-th coefficient of a general \(A\)-expansion.
INPUT:
coeff_stream(function) – corresponds to the function \(a\mapsto c_a\) where \(c_a\) is the \(a\)-th coefficient of the \(A\)-expansion;\(n\) (integer) – the exponent of the \(A\)-expansion;
i– an integer representing the index of the coefficient to compute;param– a prime power, a univariate polynomial ring over \(\mathbb{F}_q\) or the fraction field of such polynomial ring.
EXAMPLE:
sage: from drinfeld_modular_forms import coefficient_A_expansion sage: coeff_stream = lambda a: a^2 - 1 sage: coefficient_A_expansion(coeff_stream, 10, 6, 3) 2/(T^18 + 2*T^12 + 2*T^10 + T^4) sage: coeff_stream = lambda a: a^3 sage: coefficient_A_expansion(coeff_stream, 1, 7, 3) 2*T^3 + T
- drinfeld_modular_forms.expansions.coefficient_petrov_expansion(k, n, i, param)#
Return the \(i\)-th coefficient of the expansion for the form \(f_{k, n}\).
INPUT:
k– an integer representing the weight of the resulting form.n– an integer. The type of the resulting form will be congruent to this integer modulo \(q-1\).i– an integer representing the index of the coefficient to compute.param– a prime power, a univariate polynomial ring over \(\mathbb{F}_q\) or the fraction field of such polynomial ring.
EXAMPLES:
sage: from drinfeld_modular_forms import coefficient_petrov_expansion sage: q = 3 sage: coefficient_petrov_expansion(q + 1, 1, 1, q) 1 sage: coefficient_petrov_expansion(q + 1, 1, 0, q) 0 sage: coefficient_petrov_expansion(q + 1, 1, 1, q) 1 sage: coefficient_petrov_expansion(q + 1, 1, 7, q) 2*T^3 + T
- drinfeld_modular_forms.expansions.compute_A_expansion(coeff_stream, n, param, name='u')#
Return the general \(A\)-expansion defined by
\[\begin{split}c_0 + \sum_{\substack{a\in \mathbb{F}_q[T] \\ a\text{ monic}}} c_a G_n(u(az))\end{split}\]as a lazy power series.
Note that the returned power series might not always represent a Drinfeld modular form.
INPUT:
coeff_stream(function) – corresponds to the function \(a\mapsto c_a\) where \(c_a\) is the \(a\)-th coefficient of the \(A\)-expansion;\(n\) (integer) – the exponent of the \(A\)-expansion;
param– a prime power, a univariate polynomial ring over \(\mathbb{F}_q\) or the fraction field of such polynomial ring;name(string, default:'u') – represent the parameter at infinity.
EXAMPLE:
sage: from drinfeld_modular_forms import compute_A_expansion sage: c_h = lambda a: a^3 sage: compute_A_expansion(c_h, 1, 3) u + u^5 + O(u^7) sage: c_Delta = lambda a: a^6 sage: D = compute_A_expansion(c_Delta, 2, 3); D u^2 + 2*u^6 + O(u^7) sage: D[38] T^18 + 2*T^12 + 2*T^10 + T^4 + 1
sage: c = lambda a: a^5 - a + 2 sage: compute_A_expansion(c, 8, 5) 2 + ((1/(T^5+4*T))*u^4) + O(u^7)
- drinfeld_modular_forms.expansions.compute_delta_rank_2(param, name='u')#
Return the expansion of the normalized Drinfeld modular discriminant of rank 2.
Recall that for \(w\in \Omega^2(\mathbb{C}_{\infty})\) the Drinfeld modular discriminant is the leading coefficient of the Drinfeld module
\[\phi_{w} : T \mapsto T + g(z)\tau + \Delta(w)\tau^2\]corresponding to the lattice \(\Lambda_w := wA + A\). The normalized discriminant is the form \(\Delta_0(w)\) such that its first nonzero coefficient is 1.
INPUT:
param– a prime power, a univariate polynomial ring over \(\mathbb{F}_q\) or the fraction field of such polynomial ring.name(string, default:'u') – represent the parameter at infinity.
OUTPUT: a lazy power series over the base ring.
Note
We have \(\Delta_0(w) = f_{q^2 - 1, q - 1}\).
EXAMPLES:
sage: from drinfeld_modular_forms import compute_delta_rank_2 sage: q = 3 sage: D = compute_delta_rank_2(q); D u^2 + 2*u^6 + O(u^8) sage: D[38] T^18 + 2*T^12 + 2*T^10 + T^4 + 1 sage: D[56] 2*T^27 + 2*T^3 + 2*T
- drinfeld_modular_forms.expansions.compute_eisentein_series_rank_2(param, name='u')#
Return the expansion fo the normalized Drinfeld Eisenstein series of weight \(q-1\).
Recall that this form is defined by:
\[E_{q-1}(w) := \frac{T^q - T}{\tilde{\pi}^{q-1}} \sum_{(c, d)\in A^2\setminus 0} \frac{1}{(cw + d)^{q-1}}.\]where \(\tilde{\pi}\) is the Carlitz period.
INPUT:
param– a prime power, a univariate polynomial ring over \(\mathbb{F}_q\) or the fraction field of such polynomial ring.name(string, default:'u') – represent the parameter at infinity.
OUTPUT: a lazy power series over the base ring.
Note
We have \(E_{q-1} = 1 - (T^q - T)f_{q - 1, q - 1}\).
EXAMPLES:
sage: from drinfeld_modular_forms import compute_eisentein_series_rank_2 sage: q = 3 sage: E = compute_eisentein_series_rank_2(q); E 1 + ((2*T^3+T)*u^2) + O(u^7) sage: E[0:15] [1, 0, 2*T^3 + T, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*T^3 + T]
- drinfeld_modular_forms.expansions.compute_h_rank_2(param, name='u')#
Return the expansion of the function \(h\) defined by \(\Delta = h^{q-1}\) and having an expansion of the form \(u + O(u^2)\).
INPUT:
param– a prime power, a univariate polynomial ring over \(\mathbb{F}_q\) or the fraction field of such polynomial ring.name(string, default:'u') – represent the parameter at infinity.
EXAMPLES:
sage: from drinfeld_modular_forms import compute_h_rank_2 sage: q = 3 sage: h = compute_h_rank_2(q); h u + u^5 + ((2*T^3+T)*u^7) + O(u^8) sage: h[0:15] [0, 1, 0, 0, 0, 1, 0, 2*T^3 + T, 0, 1, 0, T^3 + 2*T, 0, T^6 + T^4 + T^2 + 1, 0]
- drinfeld_modular_forms.expansions.compute_petrov_expansion(k, n, param, name='u', check=True)#
Return the \(A\)-expansion of the form \(f_{k, n}\) as a lazy power series.
Recall that it is defined by:
\[\begin{split}f_{k, n} = \sum_{\substack{a\in \mathbb{F}_q[T] \\ a\text{ monic}}} a^{k - n}G_n(u(az))\end{split}\]INPUT:
k– an integer equal to the weight of the resulting form.n– an integer congruent to the type modulo \(q-1\).param– a prime power, a univariate polynomial ring over \(\mathbb{F}_q\) or the fraction field of such polynomial ring.name(string, default:'u') – represent the parameter at infinity.check(Boolean, default:True) – If this parameter is set toTrue, the code will check if \(k\) and \(n\) verify the three conditions:\(k - 2n > 0\);
\(q - 1\) divides \(k - 2n\);
\(n > p^{v_p(k - n)}\).
OUTPUT: A lazy power series.
EXAMPLES:
sage: from drinfeld_modular_forms import compute_petrov_expansion sage: q = 3 sage: D = compute_petrov_expansion(q + 1, 1, q); D u + u^5 + ((2*T^3+T)*u^7) + O(u^8)
To obtain more coefficient, one just need to take slices the series:
sage: D[0:10] # output the coefficients from i=0...9 [0, 1, 0, 0, 0, 1, 0, 2*T^3 + T, 0, 1] sage: D[39] # print the 39-th coefficient T^9 + 2*T^3 sage: D[59] # 59-th coefficient 2*T^27 + T^9 + T^3 + 2*T
- drinfeld_modular_forms.expansions.inverse_cyclotomic_polynomial(a, name='X')#
Return the polynomial \(f_a(X) = \rho_a(X^{-1}) X^{q^{\mathrm{deg}(a)}}\) where \(\rho_a\) is the Carlitz module.
This function is used in
parameter_at_infinity()INPUT:
a(polynomial) – univariate polynomial over a finite field.name(string, default:'X') – the name of the variable.
EXAMPLES:
sage: from drinfeld_modular_forms.expansions import inverse_cyclotomic_polynomial sage: A.<T> = GF(3)['T'] sage: inverse_cyclotomic_polynomial(A.one()) 1 sage: inverse_cyclotomic_polynomial(T) T*X^2 + 1 sage: inverse_cyclotomic_polynomial(T^2) T^2*X^8 + (T^3 + T)*X^6 + 1 sage: inverse_cyclotomic_polynomial(T^2 + T + 1) (T^2 + T + 1)*X^8 + (T^3 + T + 1)*X^6 + 1
- drinfeld_modular_forms.expansions.j_invariant(param, name='u')#
Return the \(j\)-invariant defined by \(j := g_1^{q+1}/g_2\).
It is normalized so that \(j = 1/u^{q - 1} + O(1)\).
INPUT:
param– a prime power, a univariate polynomial ring over \(\mathbb{F}_q\) or the fraction field of such polynomial ring.name(string, default:'u') – represent the parameter at infinity.
EXAMPLES:
sage: from drinfeld_modular_forms import j_invariant sage: q = 3 sage: j_invariant(q) 1/u^2 + (2*T^3+T) + u^2 + ((2*T^9+2*T^3+2*T)*u^4) + O(u^5) sage: q = 4 sage: j_invariant(q) 1/u^3 + (T^4+T) + O(u^4)
- drinfeld_modular_forms.expansions.parameter_at_infinity(a, name='u')#
Return the function \(u(aw)\) as a power series in \(u\).
INPUT:
a(polynomial) – univariate polynomial over a finite field.name(string, default:'u') – the name of the lazy power series ring generator.
EXAMPLES:
sage: from drinfeld_modular_forms import parameter_at_infinity sage: A.<T> = GF(3)['T'] sage: parameter_at_infinity(A.one()) u sage: parameter_at_infinity(T) u^3 + 2*T*u^5 + T^2*u^7 + 2*T^3*u^9 + O(u^10) sage: parameter_at_infinity(T^2) u^9 + ((2*T^3+2*T)*u^15) + O(u^16)