feat: add schur_index_over_center (#1802)

Author: thofma

src/AlgAss/Ramification.jl

@@ -197,6 +197,20 @@ function schur_index(A::AbstractAssociativeAlgebra{<: NumFieldElem})
   return e
 end
 
+# Can the following be done without computing the center? Not clear, since the algebra
+# might have locally different Schur indicies over places lying over the same place 
+function schur_index_over_center(A::AbstractAssociativeAlgebra)
+  C, = StructureConstantAlgebra(A)
+  B, = _as_algebra_over_center(C) 
+  return schur_index(B)
+end
+
+function schur_index_over_center(A::AbstractAssociativeAlgebra, p)
+  C, = StructureConstantAlgebra(A)
+  B, = _as_algebra_over_center(C) 
+  return schur_index(B, p)
+end
+
 ################################################################################
 #
 #  Eichler condition

src/exports.jl

@@ -824,6 +824,7 @@ export scale_row!
 export scale_row_right!
 export scales
 export schur_index
+export schur_index_over_center
 export semi_global_minimal_model
 export set_assertion_level
 export set_var!

test/AlgAss/Ramification.jl

@@ -168,4 +168,23 @@
     P = real_embeddings(K)[1]
     @test Hecke.n_positive_roots(f, P) == 21
   end
+
+  # Schur index over center
+  let
+    K, = quadratic_field(2)
+    A = Hecke.QuaternionAlgebra(K, K(-1), K(-1))
+    @test schur_index(A) == 2
+    AA, = restrict_scalars(A, QQ)
+    @test schur_index_over_center(AA) == 2
+    c = infinite_places(K)[1]
+    @test schur_index_over_center(AA, c) == 2
+
+    K, = quadratic_field(-1)
+    A = Hecke.QuaternionAlgebra(K, K(-1), K(-1))
+    @test schur_index(A) == 1
+    AA, = restrict_scalars(A, QQ)
+    @test schur_index_over_center(AA) == 1
+    c = infinite_places(K)[1]
+    @test schur_index_over_center(AA, c) == 1
+  end
 end