comment code

Author: Xavier Caruso

src/sage/rings/function_field/drinfeld_modules/charzero_drinfeld_module.py

@@ -572,18 +572,28 @@ def class_polynomial(self):
             ...
             ValueError: coefficients are not polynomials
         """
+        # The algorithm is based on the following remark:
+        # writing phi_T = g_0 + g_1*tau + ... + g_r*tau^r,
+        # if s > deg(g_i/(q^i - 1)) - 1 for all i, then the
+        # class module is equal to
+        #   H := E(Kinfty/A) / < T^(-s), T^(-s-1), ... >
+        # where E(Kinfty/A) is Kinfty/A equipped with the
+        # A-module structure coming from phi.
+
         A = self.function_ring()
         Fq = A.base_ring()
         q = Fq.cardinality()
         r = self.rank()
 
+        # We compute the bound s
         gs = self.coefficients_in_function_ring(sparse=False)
-
         s = max(gs[i].degree() // (q**i - 1) for i in range(1, r+1))
         if s == 0:
-            # small case
             return A.one()
 
+        # We compute the matrix of phi_T acting on the quotient
+        #   M := (Kinfty/A) / < T^(-s), T^(-s-1), ... >
+        # (for the standard structure of A-module!)
         M = matrix(Fq, s)
         qk = 1
         for k in range(r+1):
@@ -596,6 +606,12 @@ def class_polynomial(self):
                     M[i, j] += gs[k][e]
             qk *= q
 
+        # We compute the subspace of E(Kinfty/A) (for the twisted
+        # structure of A-module!)
+        #   V = < T^(-s), T^(-s+1), ... >
+        # It is also the phi_T-saturation of T^(-s+1) in M, i.e.
+        # the Fq-vector space generated by the phi_T^i(T^(-s+1))
+        # for i varying in NN.
         v = vector(Fq, s)
         v[s-1] = 1
         vs = [v]
@@ -605,6 +621,8 @@ def class_polynomial(self):
         V = matrix(vs)
         V.echelonize()
 
+        # We compute the action of phi_T on H = M/V
+        # as an Fq-linear map (encoded in the matrix N)
         dim = V.rank()
         pivots = V.pivots()
         j = ip = 0
@@ -613,6 +631,10 @@ def class_polynomial(self):
                 j += 1
                 ip += 1
             V[i,j] = 1
-
         N = (V * M * ~V).submatrix(dim, dim)
+
+        # The class module is now H where the action of T
+        # is given by the matrix N
+        # The class polynomial is then the characteristic
+        # polynomial of N
         return A(N.charpoly())