Remove trailing whitespace in a bunch of files

src/EllCrv/EllCrv.jl

@@ -164,11 +164,11 @@
 
 function Base.getindex(P::EllCrvPt, i::Int)
   @req 1 <= i <= 3 "Index must be 1, 2 or 3"
-  
+
   K = base_field(parent(P))
-  
+
   if is_infinite(P)
-    if i == 1 
+    if i == 1
       return zero(K)
     elseif i == 2
       return one(K)
@@ -236,14 +236,14 @@
   return elliptic_curve(QQFieldElem[QQ(z) for z in x], check = check)
 end
 
-# A constructor to create an elliptic curve from a bivariate polynomial. 
-# One can specify how to interpret the polynomial via the second and the 
+# A constructor to create an elliptic curve from a bivariate polynomial.
+# One can specify how to interpret the polynomial via the second and the
 # third argument.
 @doc raw"""
     elliptic_curve(f::MPolyRingElem, x::MPolyRingElem, y::MPolyRingElem) -> EllCrv
 
-Construct an elliptic curve from a bivariate polynomial `f` in long Weierstrass form. 
-The second and third argument specify variables of the `parent` of `f` so that 
+Construct an elliptic curve from a bivariate polynomial `f` in long Weierstrass form.
+The second and third argument specify variables of the `parent` of `f` so that
 ``f = c ⋅ (-y² + x³ - a₁ ⋅ xy + a₂ ⋅ x² - a₃ ⋅ y + a₄ ⋅ x + a₆)``.
 """
 function elliptic_curve(f::MPolyRingElem, x::MPolyRingElem, y::MPolyRingElem)
@@ -624,10 +624,10 @@
 ```
 """
 function (E::EllCrv{T})(coords::Vector{S}; check::Bool = true) where {S, T}
-  if !(2 <= length(coords) <= 3) 
+  if !(2 <= length(coords) <= 3)
     error("Points need to be given in either affine coordinates (x, y) or projective coordinates (x, y, z)")
   end
-  
+
   if length(coords) == 3
     if coords[1] == 0 && coords[3] == 0
       if coords[2] != 0
@@ -636,7 +636,7 @@
         error("The triple [0: 0: 0] does not define a point in projective space.")
       end
     end
-    coords = [coords[1]//coords[3], coords[2]//coords[3]] 
+    coords = [coords[1]//coords[3], coords[2]//coords[3]]
   end
   if S === T
     parent(coords[1]) != base_field(E) &&
@@ -1122,18 +1122,18 @@
   psi_m_univ = division_polynomial_univariate(E, m, x)[2]
   psi_mplus = division_polynomial(E, m+1, x, y)
   psi_mmin2 = division_polynomial(E, m-2, x, y)
-  
-  if p == 3 && j_invariant(E) != 0   
-    if iseven(m) 
+
+  if p == 3 && j_invariant(E) != 0
+    if iseven(m)
       num = (psi_mplus2*psi_mmin^2 - psi_mmin2*psi_mplus^2) - a1*(x* psi_m_univ^2*B6 -psi_mmin*psi_mplus)*psi_m_univ
        denom = 2*B6^2*psi_m_univ^3
     else
       num = (psi_mplus2*psi_mmin^2 - psi_mmin2*psi_mplus^2) - a1*(x* psi_m_univ^2 -psi_mmin*psi_mplus*B6)*psi_m_univ
-      denom = 4*y*psi_m_univ^3 
+      denom = 4*y*psi_m_univ^3
     end
     return num//denom
   end
-  
+
 
   num = (psi_mplus2*psi_mmin^2 - psi_mmin2*psi_mplus^2)
 

src/EllCrv/Finite.jl

@@ -35,7 +35,7 @@
 ################################################################################
 
 export hasse_interval, order, order_via_exhaustive_search, order_via_bsgs, order_via_legendre,
-       order_via_schoof, trace_of_frobenius, rand, elem_order_bsgs, is_supersingular, 
+       order_via_schoof, trace_of_frobenius, rand, elem_order_bsgs, is_supersingular,
        is_ordinary, is_probable_supersingular, supersingular_polynomial
 
 ################################################################################
@@ -924,34 +924,34 @@ Return true when the elliptic curve is supersingular. The result is proven to be
 """
 function is_supersingular(E::EllCrv{T}) where T <: FinFieldElem
   K = base_field(E)
-  
+
   p = characteristic(K)
   j = j_invariant(E)
-  
+
   if j^(p^2) != j
     return false
   end
-  
+
   if p<= 3
     return j == 0
   end
-  
+
   L = Native.GF(p, 2)
   Lx, X = polynomial_ring(L, "X")
   Lxy, Y = polynomial_ring(Lx, "Y")
   Phi2 = X^3 + Y^3 - X^2*Y^2 + 1488*(X^2*Y + Y^2*X) - 162000*(X^2 + Y^2) + 40773375*X*Y + 8748000000*(X + Y) - 157464000000000
-  
+
   jL = _embed_into_p2(j, L)
-  
+
   js = roots(Phi2(jL))
-  
+
   if length(js) < 3
     return false
   end
-  
+
   newjs = [jL, jL, jL]
   f = elem_type(Lx)[zero(Lx), zero(Lx), zero(Lx)]
-  
+
   m = nbits(p) - 1
   for k in (1 : m)
     for i in (1 : 3)
@@ -1016,15 +1016,15 @@ function is_probable_supersingular(E::EllCrv{T}) where T <: FinFieldElem
   j = j_invariant(E)
   K = base_field(E)
   p = characteristic(K)
-  
+
   local degj::Int
 
   if degree(K) == 1
     degj = 1
   else
     degj = degree(minpoly(j))
   end
-  
+
   if degj == 1
     return monte_carlo_test(E, p+1)
   elseif degj == 2
@@ -1036,14 +1036,14 @@ end
 
 function monte_carlo_test(E, n)
   E_O = infinity(E)
-  
+
   for i in (1:10)
     P = rand(E)
     if n*P != E_O
       return false
     end
   end
-  
+
   return true
 end
 
@@ -1060,7 +1060,7 @@ function supersingular_polynomial(p::IntegerUnion)
   if p < 3
     return J
   end
-  
+
   m = divexact((p-1), 2)
   KXT, (X, T) = polynomial_ring(K, ["X", "T"])
   H = sum([binomial(m, i)^2 *T^i for i in (0:m)])
@@ -1144,7 +1144,7 @@ Return a list of generators of the group of rational points on $E$.
 
 # Examples
 
-```jldoctest; filter = r"Point.*" 
+```jldoctest; filter = r"Point.*"
 julia> E = elliptic_curve(GF(101, 2), [1, 2]);
 
 julia> gens(E)

src/EllCrv/FormalGroupLaw.jl

@@ -19,17 +19,17 @@ function formal_w(E::EllCrv, prec::Int = 20)
   k = base_field(E)
   kz, z = laurent_series_ring(k, prec, "z")
   kzw, w = polynomial_ring(kz, "w")
-  
+
   a1, a2, a3, a4, a6 = a_invars(E)
-  
+
   f = z^3 + a1*z*w + a2*z^2*w + a3*w^2 + a4*z*w^2 + a6*w^3
   result = z^3 + a1*z*w + a2*z^2*w + a3*w^2 + a4*z*w^2 + a6*w^3
-  
+
   for i in (1:prec)
     result = truncate(f(result), div(10,3))
   end
   return result(0)
-  
+
 end
 
 @doc raw"""
@@ -69,7 +69,7 @@ function formal_differential_form(E::EllCrv, prec::Int = 20)
   x = formal_x(E, prec + 1)
   y = formal_y(E, prec + 1)
   dx = derivative(x)
-  
+
   return truncate(dx//(2*y + a1*x + a3), prec)
 end
 
@@ -87,29 +87,29 @@ end
   k = base_field(E)
   a1, a2, a3, a4, a6 = a_invars(E)
   ktt, (z1, z2) = power_series_ring(k, prec + 1, ["z1", "z2"])
-  
+
   #We now compute the slope lambda of the addition in the formal group law
   #Following Silverman
   #A_n-3 = w[n] for n in 3 to infinity
-  #(z1^n - z2^n)//(z1 - z2) = z1^(n-1) + z1^(n-2)*z2 +.... +z2^(n-2) 
-  #lambda = sum A_n-3*(z1^n - z2^n)//(z1 - z2) 
-  
+  #(z1^n - z2^n)//(z1 - z2) = z1^(n-1) + z1^(n-2)*z2 +.... +z2^(n-2)
+  #lambda = sum A_n-3*(z1^n - z2^n)//(z1 - z2)
+
   w = formal_w(E, prec + 1)
-  
+
   lambda = sum([coeff(w, n)*sum([z1^m * z2^(n-m-1) for m in (0:n-1)]) for n in (3:prec +1)])
-  
+
   #Needs to simply be evaluating
   wz1 = sum([coeff(w, i)*z1^i for i in (0:prec +1)])
-  
+
   nu = wz1 - lambda*z1
-  
+
   z3 = -z1 - z2 - divexact(a1*lambda + a3*lambda^2 + a2*nu + 2*a4*lambda*nu + 3*a6*lambda^2*nu, 1 + a2*lambda + a4*lambda^2 + a6*lambda^3)
-  
+
   inv = formal_inverse(E, prec)
-  
+
   #Needs to simply be evaluating
   result = sum([coeff(inv, i)*z3^i for i in (0:prec +1)])
-  
+
   #Sage and Magma truncate to O(z1, z2)^20 instead of O(z1)^20 + O(z2)^20 like we do
   return result
 end
@@ -138,11 +138,11 @@ function formal_isogeny(phi::Isogeny, prec::Int = 20)
   E = domain(phi)
   x = formal_x(E, prec)
   y = formal_y(E, prec)
-  
+
   Rz = parent(x)
-  
+
   maps = rational_maps(phi)
-  
+
   fnum = change_base_ring(Rz, numerator(maps[1]))
   fdenom = change_base_ring(Rz, denominator(maps[1]))
   gnum = change_base_ring(Rz, numerator(maps[2]))