Initial implementation of quasimodular forms

src/sage/modular/all.py

@@ -40,3 +40,5 @@
 from .btquotients.all import *
 
 from .pollack_stevens.all import *
+
+from .quasimodform.all import *

src/sage/modular/quasimodform/__init__.py

@@ -0,0 +1 @@
+from . import all

src/sage/modular/quasimodform/all.py

@@ -0,0 +1 @@
+from .quasimodform import QuasiModularFormsRing

src/sage/modular/quasimodform/quasimodform.py

@@ -0,0 +1,228 @@
+r"""
+Graded quasi-modular forms ring
+
+TODO: add more info
+
+.. NOTE:
+
+    Currently, all the methods are implemented only for the full modular groups. Congruence subgroups
+    of higher level are not yet supported
+
+AUTHORS:
+
+- DAVID AYOTTE (2021-03-18): initial version
+
+"""
+
+# ****************************************************************************
+#       Copyright (C) 2021 DAVID AYOTTE
+#
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
+
+
+from sage.modular.modform.eis_series import eisenstein_series_qexp
+from sage.modular.arithgroup.all import Gamma0, is_CongruenceSubgroup
+from sage.rings.all import Integer, QQ, ZZ
+from sage.structure.sage_object  import SageObject
+
+class QuasiModularFormsRing(SageObject):
+    def __init__(self, group=1, base_ring=QQ):
+        r"""
+        The graded ring of quasimodular forms for the full modular group `{\rm SL}_2(\ZZ)`, with
+        coefficients in a ring.
+
+        INPUT:
+
+        - ``group`` (default: `{\rm SL}_2(\ZZ)`) -- a congruence subgroup of `{\rm SL}_2(\ZZ)`, or a
+          positive integer `N` (interpreted as `\Gamma_0(N)`)
+
+        - ``base_ring`` (ring, default: `\QQ`) -- a base ring, which should be
+          `\QQ`, `\ZZ`, or the integers mod `p` for some prime `p`.
+
+        EXAMPLES::
+
+            sage: M = QuasiModularFormsRing(); M
+            Ring of quasimodular forms for Modular Group SL(2,Z) with coefficients in Rational Field
+            sage: B = M.generators(); B
+            [(2,
+            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 - 288*q^6 - 192*q^7 - 360*q^8 - 312*q^9 + O(q^10)),
+            (4,
+            1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + 60480*q^6 + 82560*q^7 + 140400*q^8 + 181680*q^9 + O(q^10)),
+            (6,
+            1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 - 4058208*q^6 - 8471232*q^7 - 17047800*q^8 - 29883672*q^9 + O(q^10))]
+            sage: P = B[0][1]; Q = B[1][1]
+            sage: D = M.differentiation_operator
+            sage: D(P)
+            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 - 1728*q^6 - 1344*q^7 - 2880*q^8 - 2808*q^9 + O(q^10)
+            sage: (P^2 - Q)/12
+            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 - 1728*q^6 - 1344*q^7 - 2880*q^8 - 2808*q^9 + O(q^10)
+            sage: M = QuasiModularFormsRing(1, Integers(5)); M
+            Ring of quasimodular forms for Modular Group SL(2,Z) with coefficients in Ring of integers modulo 5
+            sage: B = M.generators(); B
+            [(2, 1 + q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + 2*q^6 + 3*q^7 + 3*q^9 + O(q^10)),
+            (4, 1 + O(q^10)),
+            (6, 1 + q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + 2*q^6 + 3*q^7 + 3*q^9 + O(q^10))]
+
+        .. TESTS:
+
+            sage: M = QuasiModularFormsRing(1)
+            sage: M.group()
+            Modular Group SL(2,Z)
+            sage: M.base_ring()
+            Rational Field
+            sage: M = QuasiModularFormsRing(1, ZZ)
+            sage: M.base_ring()
+            Integer Ring
+            sage: M = QuasiModularFormsRing(1, Integers(5))
+            sage: M.base_ring()
+            Ring of integers modulo 5
+            sage: QuasiModularFormsRing(2)
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: The space of quasimodular forms for higher levels are not yet implemented
+            sage: QuasiModularFormsRing(Integers(5))
+            Traceback (most recent call last):
+            ...
+            ValueError: Group (=Ring of integers modulo 5) should be a congruence subgroup           
+
+
+        """
+        if isinstance(group, (int, Integer)):
+            if group>1:
+                raise NotImplementedError("The space of quasimodular forms for higher levels are not yet implemented")
+            group = Gamma0(1)
+        elif not is_CongruenceSubgroup(group):
+            raise ValueError("Group (=%s) should be a congruence subgroup" % group)
+        elif group is not Gamma0(1):
+            raise NotImplementedError("The space of quasimodular forms for higher levels are not yet implemented")
+
+        self.__group = group
+        self.__base_ring = base_ring
+
+    def group(self):
+        r"""
+        Return the congruence subgroup for which this is the ring of quasimodular forms.
+
+        EXAMPLES::
+
+            sage: M = QuasiModularFormsRing(1)
+            sage: M.group() is SL2Z
+            True
+            sage: M = QuasiModularFormsRing(Gamma0(1)); M
+            Ring of quasimodular forms for Modular Group SL(2,Z) with coefficients in Rational Field
+            
+        Higher level congruence subgroups are not yet implemented::
+
+            sage: QuasiModularFormsRing(2)
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: The space of quasimodular forms for higher levels are not yet implemented
+
+        """
+        return self.__group
+
+    def base_ring(self):
+        r"""
+        Return the coefficient ring of this quasimodular forms ring.
+
+        EXAMPLES::
+
+            sage: QuasiModularFormsRing(1).base_ring()
+            Rational Field
+            sage: QuasiModularFormsRing(1, base_ring = ZZ).base_ring()
+            Integer Ring
+            sage: QuasiModularFormsRing(1, base_ring = Integers(5)).base_ring()
+            Ring of integers modulo 5
+        """
+        return self.__base_ring
+
+    def _repr_(self):
+        r"""
+        String representation of self.
+
+        EXAMPLES::
+
+            sage: QuasiModularFormsRing(1)._repr_()
+            'Ring of quasimodular forms for Modular Group SL(2,Z) with coefficients in Rational Field'
+            sage: QuasiModularFormsRing(1, base_ring=Integers(13))._repr_()
+            'Ring of quasimodular forms for Modular Group SL(2,Z) with coefficients in Ring of integers modulo 13'
+        """
+        return "Ring of quasimodular forms for %s with coefficients in %s" % (self.group(), self.base_ring())
+
+    def generators(self, prec=10):
+        r"""
+        If `R` is the base ring of self, then this method returns a set of
+        quasimodular forms which generate the `R`-algebra of all quasimodular forms.
+
+        INPUT:
+
+        - ``prec`` (integer, default: 10) -- return `q`-expansions to this
+          precision
+
+        OUPUT:
+
+        a list of pairs (k, f), where f is the q-expansion to precision
+        ``prec`` of a quasimodular form of weight k. For the full modular group, these
+        forms are precisely the normalized eisenstein series of weight 2, 4 and 6 respectively.
+
+        EXAMPLES::
+
+            sage: M = QuasiModularFormsRing(1)
+            sage: M.generators()
+            [(2,
+            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 - 288*q^6 - 192*q^7 - 360*q^8 - 312*q^9 + O(q^10)),
+            (4,
+            1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + 60480*q^6 + 82560*q^7 + 140400*q^8 + 181680*q^9 + O(q^10)),
+            (6,
+            1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 - 4058208*q^6 - 8471232*q^7 - 17047800*q^8 - 29883672*q^9 + O(q^10))]
+            sage: QuasiModularFormsRing(1, Integers(17)).generators(prec=6)
+            [(2, 1 + 10*q + 13*q^2 + 6*q^3 + 2*q^4 + 9*q^5 + O(q^6)),
+            (4, 1 + 2*q + q^2 + 5*q^3 + 10*q^4 + 14*q^5 + O(q^6)),
+            (6, 1 + 6*q + 11*q^2 + 2*q^3 + q^4 + 5*q^5 + O(q^6))]
+        """
+        E2 = eisenstein_series_qexp(2, prec=prec, K=self.base_ring(), normalization='constant')
+        E4 = eisenstein_series_qexp(4, prec=prec, K=self.base_ring(), normalization='constant')
+        E6 = eisenstein_series_qexp(6, prec=prec, K=self.base_ring(), normalization='constant')
+        return [(2, E2), (4, E4), (6, E6)]
+
+    def differentiation_operator(self, f):
+        r"""
+        Compute the formal derivative `q\frac{d}{dq}` of the q-expansion of a quasimodular form `f`
+
+        INPUT:
+
+        - ``f`` -- a power serie in corresponding to the q-expansion of a quasimodular form.
+
+        OUTPUT:
+
+        The power serie `q\frac{d}{dq}(f)`
+
+        EXAMPLES::
+
+            sage: M = QuasiModularFormsRing()
+            sage: D = M.differentiation_operator
+            sage: B = M.generators()
+            sage: P = B[0][1]; Q = B[1][1]; R = B[2][1]
+            sage: D(P)
+            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 - 1728*q^6 - 1344*q^7 - 2880*q^8 - 2808*q^9 + O(q^10)
+            sage: (P^2 - Q)/12
+            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 - 1728*q^6 - 1344*q^7 - 2880*q^8 - 2808*q^9 + O(q^10)
+            sage: D(Q)
+            240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*q^5 + 362880*q^6 + 577920*q^7 + 1123200*q^8 + 1635120*q^9 + O(q^10)
+            sage: (P*Q - R)/3
+            240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*q^5 + 362880*q^6 + 577920*q^7 + 1123200*q^8 + 1635120*q^9 + O(q^10)
+            sage: D(R)
+            -504*q - 33264*q^2 - 368928*q^3 - 2130912*q^4 - 7877520*q^5 - 24349248*q^6 - 59298624*q^7 - 136382400*q^8 - 268953048*q^9 + O(q^10)
+            sage: (P*R - Q^2)/2
+            -504*q - 33264*q^2 - 368928*q^3 - 2130912*q^4 - 7877520*q^5 - 24349248*q^6 - 59298624*q^7 - 136382400*q^8 - 268953048*q^9 + O(q^10)
+        """
+        q = f.parent().gen()
+        return q*f.derivative()
+
+
+