establish interface for instantiated classical modular polynomials

Author: yyyyx4

src/doc/en/reference/arithmetic_curves/index.rst

@@ -25,6 +25,7 @@ Maps between them
    sage/schemes/elliptic_curves/hom_scalar
    sage/schemes/elliptic_curves/hom_frobenius
    sage/schemes/elliptic_curves/isogeny_small_degree
+   sage/schemes/elliptic_curves/mod_poly
 
 
 Elliptic curves over number fields

src/sage/schemes/elliptic_curves/all.py

@@ -40,4 +40,6 @@
 
 from .ell_curve_isogeny import EllipticCurveIsogeny, isogeny_codomain_from_kernel
 
+from .mod_poly import classical_modular_polynomial
+
 from .heegner import heegner_points, heegner_point

src/sage/schemes/elliptic_curves/mod_poly.py

@@ -0,0 +1,142 @@
+r"""
+Modular polynomials for elliptic curves
+
+For a positive integer `\ell`, the classical modular polynomial
+`\Phi_\ell\in\ZZ[X,Y]` is characterized by the property that its
+zero set is exactly the set of pairs of `j`-invariants connected
+by a cyclic `\ell`-isogeny.
+
+AUTHORS:
+
+- Lorenz Panny (2023)
+"""
+
+from sage.misc.cachefunc import cached_function
+from sage.structure.parent import Parent
+from sage.structure.element import parent, FieldElement
+
+from sage.rings.integer_ring import ZZ
+from sage.rings.polynomial.polynomial_ring import polygen, polygens
+
+from sage.libs.pari import pari
+from cypari2.handle_error import PariError
+
+from sage.databases.db_modular_polynomials import ClassicalModularPolynomialDatabase
+_db = ClassicalModularPolynomialDatabase()
+
+_cache = dict()
+
+def classical_modular_polynomial(l, j=None):
+    r"""
+    Return the classical modular polynomial `\Phi_\ell`, either as a
+    "generic" bivariate polynomial over `\ZZ`, or as an "instantiated"
+    modular polynomial where one variable has been replaced by the
+    given `j`-invariant.
+
+    Generic polynomials are cached up to a certain size of `\ell`,
+    which significantly accelerates subsequent invocations with the
+    same `\ell`. The default bound is `\ell \leq 150`, which can be
+    adjusted by setting ``classical_modular_polynomial.cache_bound``
+    to a different value. Beware that modular polynomials are very
+    large and the amount of memory consumed by the cache will grow
+    rapidly when the bound is set to a large value.
+
+    INPUT:
+
+    - ``l`` -- positive integer.
+
+    - ``j`` -- either ``None`` or a ring element.
+
+      - If ``None`` is given, the original modular polynomial
+        is returned as an element of `\ZZ[X,Y]`.
+
+      - If a ring element `j \in R` is given, the evaluation
+        `\Phi_\ell(j,Y)` is returned as an element of the
+        univariate polynomial ring `R[Y]`.
+
+    ALGORITHMS:
+
+    - The Kohel database
+      :class:`~sage.databases.db_modular_polynomials.ClassicalModularPolynomialDatabase`
+
+    - :pari:`polmodular`
+
+    EXAMPLES::
+
+        sage: classical_modular_polynomial(2)
+        -X^2*Y^2 + X^3 + 1488*X^2*Y + 1488*X*Y^2 + Y^3 - 162000*X^2 + 40773375*X*Y - 162000*Y^2 + 8748000000*X + 8748000000*Y - 157464000000000
+        sage: classical_modular_polynomial(3, Zmod(10))
+        9*X^3*Y^3 + 2*X^3*Y^2 + 2*X^2*Y^3 + X^4 + 4*X^3*Y + 6*X^2*Y^2 + 4*X*Y^3 + Y^4
+        sage: j = Mod(1728, 419)
+        sage: classical_modular_polynomial(3, j)
+        Y^4 + 230*Y^3 + 84*Y^2 + 118*Y + 329
+
+    TESTS::
+
+        sage: q = random_prime(50)^randrange(1,4)
+        sage: j = GF(q).random_element()
+        sage: l = random_prime(50)
+        sage: Y = polygen(parent(j), 'Y')
+        sage: classical_modular_polynomial(l,j) == classical_modular_polynomial(l)(j,Y)
+        True
+    """
+    l = ZZ(l)
+
+    if j is None:
+        # We are supposed to return the generic modular polynomial. First
+        # check if it is already in the cache, then check the database,
+        # finally compute it using PARI.
+        try:
+            return _cache[l]
+        except KeyError:
+            pass
+
+        try:
+            Phi = ZZ['X,Y'](_db[l])
+        except ValueError:
+            try:
+                pari_Phi = pari.polmodular(l)
+            except PariError:
+                raise NotImplementedError('modular polynomial is not in database and computing it on the fly is not yet implemented')
+            d = {(i,j): c for i,f in enumerate(pari_Phi) for j,c in enumerate(f)}
+            Phi = ZZ['X,Y'](d)
+
+        if l <= classical_modular_polynomial.cache_bound:
+            _cache[l] = Phi
+
+        return Phi
+
+    R,Y = parent(j)['Y'].objgen()
+
+    # If the generic polynomial is in the cache or the database, evaluating
+    # it directly should always be faster than recomputing it from scratch.
+    try:
+        Phi = _cache[l]
+    except KeyError:
+        pass
+    else:
+        return Phi(j, Y)
+    try:
+        Phi = _db[l]
+    except ValueError:
+        pass
+    else:
+        if l <= classical_modular_polynomial.cache_bound:
+            _cache[l] = ZZ['X,Y'](Phi)
+        return Phi(j, Y)
+
+    # Now try to get the instantiated modular polynomial directly from PARI.
+    # This should be slightly more efficient (in particular regarding memory
+    # usage) than computing and evaluating the generic modular polynomial.
+    try:
+        pari_Phi = pari.polmodular(l, 0, j)
+    except PariError:
+        pass
+    else:
+        return R(pari_Phi)
+
+    # Nothing worked. Fall back to computing the generic modular polynomial
+    # and simply evaluating it.
+    return classical_modular_polynomial(l)(j, Y)
+
+classical_modular_polynomial.cache_bound = 150