establish interface for instantiated classical modular polynomials

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 +1,94 @@
+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.
+
+The functions in this file compute *evaluations* `\Phi_\ell(j,Y)`
+of `\Phi_\ell`, univariate polynomials whose roots are exactly
+the `j`-invariants of `\ell`-isogeny neighbours of the given `j`.
+
+Currently, the only supported algorithm is a simple database lookup
+followed by evaluation. Nevertheless, it makes sense to use the
+interface provided by :func:`classical_modular_polynomial` already,
+as better algorithms may be implemented and automatically selected
+in the future.
+
+AUTHORS:
+
+- Lorenz Panny (2023)
+"""
+
+from sage.structure.parent import Parent
+from sage.structure.element import parent
+from sage.rings.integer_ring import ZZ
+from sage.rings.polynomial.polynomial_ring import polygen, polygens
+from sage.databases.db_modular_polynomials import ClassicalModularPolynomialDatabase
+
+def classical_modular_polynomial(l, inst=None):
+    r"""
+    Return the classical modular polynomial `\Phi_\ell`,
+    either as a bivariate polynomial over `\ZZ`, a bivariate
+    polynomial over another ring, or a univariate polynomial
+    which is the result of evaluating `\Phi_\ell` at a given
+    `j`-invariant.
+
+    INPUT:
+
+    - ``l`` -- positive integer.
+
+    - ``inst`` -- either ``None``, a ring, or a ring element.
+
+      - If ``None`` is given, the original modular polynomial
+        is returned as an element of `\ZZ[X,Y]`.
+
+      - If a ring `R` is given, the base change of the modular
+        polynomial to `R[X,Y]` is returned.
+
+      - 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]`.
+
+    The Kohel database
+    :class:`~sage.databases.db_modular_polynomials.ClassicalModularPolynomialDatabase`
+    may need to be installed.
+
+    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)
+
+    db = ClassicalModularPolynomialDatabase()
+    try:
+        Phi = db[l]
+    except ValueError:
+        raise NotImplementedError('modular polynomial is not in database and computing it on the fly is not yet implemented')
+
+    if inst is None:
+        X,Y = polygen(ZZ, 'X,Y')
+    elif isinstance(inst, Parent):
+        R = inst
+        X,Y = polygen(R, 'X,Y')
+    else:
+        j = inst
+        X,Y = j, polygen(parent(j), 'Y')
+
+    return Phi(X, Y)