Add is_modular(L, p) for ZZLat (#1081)

* Add is_modular(L, p) for ZZLat

Co-authored-by: Stevell Muller <[email protected]>

src/QuadForm/Lattices.jl

@@ -834,11 +834,11 @@ function is_modular(L::AbstractLat)
 end
 
 @doc raw"""
-    is_modular(L::AbstractLat, p::NfOrdIdl) -> Bool, Int
+    is_modular(L::AbstractLat, p) -> Bool, Int
 
-Return whether the completion $L_{p}$ of the lattice `L` at the prime ideal `p`
-is modular. If it is the case the second returned value is an integer `v` such
-that $L_{p}$ is $p^v$-modular.
+Return whether the completion $L_{p}$ of the lattice `L` at the prime ideal or
+integer `p` is modular. If it is the case the second returned value is an
+integer `v` such that $L_{p}$ is $p^v$-modular.
 """
 is_modular(::AbstractLat, p)
 

src/QuadForm/Quad/GenusRep.jl

@@ -71,13 +71,13 @@ end
 ################################################################################
 
 @doc raw"""
-    genus_representatives(L::QuadLat; max = inf, use_auto = false)
+    genus_representatives(L::QuadLat; max = inf, use_auto = true)
                                                         -> Vector{QuadLat}
 
 Computes representatives for the isometry classes in the genus of $L$.
 
 At most `max` representatives are returned. The use of automorphims can be
-disabled by
+disabled by setting `use_auto = false`.
 """
 function genus_representatives(L::QuadLat; max = inf, use_auto = true, use_mass = true)
   # Otherwise the isomorphism to the class group fails, cf. §102 in O'Meara.

src/QuadForm/Quad/ZLattices.jl

@@ -919,6 +919,22 @@ Return the number of (positive, zero, negative) inertia of `L`.
 """
 signature_tuple(L::ZZLat) = signature_tuple(rational_span(L))
 
+################################################################################
+#
+#  Modularity
+#
+################################################################################
+
+function is_modular(L::ZZLat, p::IntegerUnion)
+  a = scale(L)
+  v = valuation(a, p)
+  if v * rank(L) == valuation(volume(L), p)
+    return true, v
+  else
+    return false, 0
+  end
+end
+
 ################################################################################
 #
 #  Local basis matrix
@@ -1080,7 +1096,6 @@ function _maximal_integral_lattice(L::ZZLat)
   return _to_ZLat(M, V = ambient_space(L))
 end
 
-
 @doc raw"""
     maximal_even_lattice(L::ZZLat, p) -> ZZLat
 

test/QuadForm/Quad/ZLattices.jl

@@ -725,3 +725,12 @@ end
   L = @inferred quadratic_lattice(QQ, [B[i,:] for i in 1:nrows(B)], gram = gram_matrix(E8))
   @test genus(L) == genus(E8)
 end
+
+let
+  L = integer_lattice(; gram = ZZ[3 0; 0 1])
+  fl, v = @inferred is_modular(L, 3)
+  @test !fl
+  L = integer_lattice(; gram = ZZ[3 0; 0 3])
+  fl, v = @inferred is_modular(L, 3)
+  @test fl && v == 1
+end