gh-37613: Fix hyperelliptic curve dynamic class construction to allow proper method inheritance
    
While working on some hyperelliptic code, I realised the dynamic class
construction had an issue with inheritance for the genus two classes and
certain methods which should have been available were not.

This is discussed in more detail in the issue
#37612

I do not know a *good* fix for this, but I have found a fix which at
least allows the functions supposed to be accessed to be accessed.

I have added a small doctest to ensure that the method called for genus
two curves is bound to the correct class.

## Overview of the fix

The original class was constructed as:

```py
    cls = dynamic_class(class_name,
                        tuple(superclass),
                        cls=HyperellipticCurve_generic,
                        doccls=HyperellipticCurve)
```

Where `superclass` is either an empty list, a list with a single child
from:

- `HyperellipticCurve_finite_field`
- `HyperellipticCurve_rational_field`
- `HyperellipticCurve_padic_field`
- `HyperellipticCurve_g2`

Notice that all four of these classes are children of
`HyperellipticCurve_generic`.

Or, in the case of the base field being one of the above AND the curve
being genus two this list is of the form:

```
[HyperellipticCurve_XXX_field, HyperellipticCurve_g2]
```

In this case, I found that the dynamic class insertion into `cls` meant
that the methods shared by one of the "`superclass`" (probably should be
called base classes?) and `cls` itself would call the method from `cls`
rather than the superclass and so all specialised functions were
unavailable, making the genus two optimisations redundant.

In my new fix, if `superclass` has a length of zero, I set `cls` to be
`HyperellipticCurve_generic`, otherwise `cls` is `None`.

This seems to work, but I don't know if this is good practice.

Fixes #37612
    
URL: #37613
Reported by: Giacomo Pope
Reviewer(s): Giacomo Pope, Kwankyu Lee

Author: Release Manager

src/sage/schemes/hyperelliptic_curves/constructor.py

@@ -92,11 +92,11 @@ def HyperellipticCurve(f, h=0, names=None, PP=None, check_squarefree=True):
         sage: HyperellipticCurve(x^8 + 1, x)
         Traceback (most recent call last):
         ...
-        ValueError: Not a hyperelliptic curve: highly singular at infinity.
+        ValueError: not a hyperelliptic curve: highly singular at infinity
         sage: HyperellipticCurve(x^8 + x^7 + 1, x^4)
         Traceback (most recent call last):
         ...
-        ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
+        ValueError: not a hyperelliptic curve: singularity in the provided affine patch
 
         sage: F.<t> = PowerSeriesRing(FiniteField(2))
         sage: P.<x> = PolynomialRing(FractionField(F))
@@ -128,7 +128,7 @@ def HyperellipticCurve(f, h=0, names=None, PP=None, check_squarefree=True):
         sage: HyperellipticCurve((x^3-x+2)^2*(x^6-1))
         Traceback (most recent call last):
         ...
-        ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
+        ValueError: not a hyperelliptic curve: singularity in the provided affine patch
 
         sage: HyperellipticCurve((x^3-x+2)^2*(x^6-1), check_squarefree=False)
         Hyperelliptic Curve over Finite Field of size 7 defined by
@@ -147,7 +147,7 @@ def HyperellipticCurve(f, h=0, names=None, PP=None, check_squarefree=True):
         sage: HyperellipticCurve(f, h)
         Traceback (most recent call last):
         ...
-        ValueError: Not a hyperelliptic curve: highly singular at infinity.
+        ValueError: not a hyperelliptic curve: highly singular at infinity
 
         sage: HyperellipticCurve(F)
         Hyperelliptic Curve over Rational Field defined by y^2 = x^6 + 1
@@ -160,7 +160,7 @@ def HyperellipticCurve(f, h=0, names=None, PP=None, check_squarefree=True):
         sage: HyperellipticCurve(x^5 + t)
         Traceback (most recent call last):
         ...
-        ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
+        ValueError: not a hyperelliptic curve: singularity in the provided affine patch
 
     Input with integer coefficients creates objects with the integers
     as base ring, but only checks smoothness over `\QQ`, not over Spec(`\ZZ`).
@@ -203,59 +203,72 @@ def HyperellipticCurve(f, h=0, names=None, PP=None, check_squarefree=True):
     """
     # F is the discriminant; use this for the type check
     # rather than f and h, one of which might be constant.
-    F = h**2 + 4*f
+    F = h**2 + 4 * f
     if not isinstance(F, Polynomial):
-        raise TypeError("Arguments f (= %s) and h (= %s) must be polynomials" % (f, h))
+        raise TypeError(f"arguments {f = } and {h = } must be polynomials")
     P = F.parent()
     f = P(f)
     h = P(h)
     df = f.degree()
-    dh_2 = 2*h.degree()
+    dh_2 = 2 * h.degree()
     if dh_2 < df:
-        g = (df-1)//2
+        g = (df - 1) // 2
     else:
-        g = (dh_2-1)//2
+        g = (dh_2 - 1) // 2
     if check_squarefree:
         # Assuming we are working over a field, this checks that after
         # resolving the singularity at infinity, we get a smooth double cover
         # of P^1.
         if P(2) == 0:
             # characteristic 2
             if h == 0:
-                raise ValueError("In characteristic 2, argument h (= %s) must be non-zero." % h)
-            if h[g+1] == 0 and f[2*g+1]**2 == f[2*g+2]*h[g]**2:
-                raise ValueError("Not a hyperelliptic curve: "
-                                 "highly singular at infinity.")
-            should_be_coprime = [h, f*h.derivative()**2+f.derivative()**2]
+                raise ValueError(
+                    f"for characteristic 2, argument {h = } must be non-zero"
+                )
+            if h[g + 1] == 0 and f[2 * g + 1] ** 2 == f[2 * g + 2] * h[g] ** 2:
+                raise ValueError(
+                    "not a hyperelliptic curve: highly singular at infinity"
+                )
+            should_be_coprime = [h, f * h.derivative() ** 2 + f.derivative() ** 2]
         else:
             # characteristic not 2
-            if F.degree() not in [2*g+1, 2*g+2]:
-                raise ValueError("Not a hyperelliptic curve: "
-                                 "highly singular at infinity.")
+            if F.degree() not in [2 * g + 1, 2 * g + 2]:
+                raise ValueError(
+                    "not a hyperelliptic curve: highly singular at infinity"
+                )
             should_be_coprime = [F, F.derivative()]
         try:
             smooth = should_be_coprime[0].gcd(should_be_coprime[1]).degree() == 0
         except (AttributeError, NotImplementedError, TypeError):
             try:
                 smooth = should_be_coprime[0].resultant(should_be_coprime[1]) != 0
             except (AttributeError, NotImplementedError, TypeError):
-                raise NotImplementedError("Cannot determine whether "
-                      "polynomials %s have a common root. Use "
-                      "check_squarefree=False to skip this check." %
-                      should_be_coprime)
+                raise NotImplementedError(
+                    "cannot determine whether "
+                    f"polynomials {should_be_coprime} have a common root, use "
+                    "check_squarefree=False to skip this check"
+                )
         if not smooth:
-            raise ValueError("Not a hyperelliptic curve: "
-                             "singularity in the provided affine patch.")
+            raise ValueError(
+                "not a hyperelliptic curve: singularity in the provided affine patch"
+            )
     R = P.base_ring()
     PP = ProjectiveSpace(2, R)
     if names is None:
         names = ["x", "y"]
 
-    superclass = []
+    bases = []
     cls_name = ["HyperellipticCurve"]
 
+    # For certain genus we specialise to subclasses with
+    # optimised methods
     genus_classes = {2: HyperellipticCurve_g2}
+    if g in genus_classes:
+        bases.append(genus_classes[g])
+        cls_name.append(f"g{g}")
 
+    # For certain base fields, we specialise to subclasses
+    # with special case methods
     def is_FiniteField(x):
         return isinstance(x, FiniteField)
 
@@ -265,19 +278,22 @@ def is_pAdicField(x):
     fields = [
         ("FiniteField", is_FiniteField, HyperellipticCurve_finite_field),
         ("RationalField", is_RationalField, HyperellipticCurve_rational_field),
-        ("pAdicField", is_pAdicField, HyperellipticCurve_padic_field)]
-
-    if g in genus_classes:
-        superclass.append(genus_classes[g])
-        cls_name.append("g%s" % g)
+        ("pAdicField", is_pAdicField, HyperellipticCurve_padic_field),
+    ]
 
     for name, test, cls in fields:
         if test(R):
-            superclass.append(cls)
+            bases.append(cls)
             cls_name.append(name)
             break
 
+    # If no specialised subclasses are identified, we simply use the
+    # generic class in the class construction
+    if not bases:
+        bases = [HyperellipticCurve_generic]
+
+    # Dynamically build a class from multiple inheritance. Note that
+    # all classes we select from are subclasses of HyperellipticCurve_generic
     class_name = "_".join(cls_name)
-    cls = dynamic_class(class_name, tuple(superclass),
-                        HyperellipticCurve_generic, doccls=HyperellipticCurve)
+    cls = dynamic_class(class_name, tuple(bases), doccls=HyperellipticCurve)
     return cls(PP, f, h, names=names, genus=g)

src/sage/schemes/hyperelliptic_curves/hyperelliptic_g2.py

@@ -45,6 +45,15 @@ def jacobian(self):
             sage: f = x^5 - x^4 + 3
             sage: HyperellipticCurve(f).jacobian()
             Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^5 - x^4 + 3
+
+        TESTS:
+
+        Ensure that :issue:`37612` is fixed::
+
+            sage: R.<x> = QQ[]
+            sage: f = x^5 - x^4 + 3
+            sage: type(HyperellipticCurve(f).jacobian())
+            <class 'sage.schemes.hyperelliptic_curves.jacobian_g2.HyperellipticJacobian_g2_with_category'>
         """
         return jacobian_g2.HyperellipticJacobian_g2(self)