partial progress towards recovering char poly

Author: stevehuang235

src/CharPolyFrob.jl

@@ -0,0 +1,94 @@
+"""
+    prec_vec(polygon, relative_precision)
+
+returns the vector by adding the height of the Hodge polygon and appropriate relative precision for each position
+
+INPUTS: 
+* "polygon" -- A SlopesPolygon struct describing a polygon whose values describe the
+divisibility of the roots.
+* "relative_precision" -- list, list of relative precisions, can be computed by "calculate_relative_precision"
+"""
+function prec_vec(polygon, relative_precision)
+    vals = Int.(polygon.values)
+    hodge_numbers = polygon.slopelengths
+    prec_vec = reverse(vals)
+    i = 1
+    num_term = 1 
+    for j in 1:length(hodge_numbers)
+        num_term = num_term + hodge_numbers[j]
+        while i < num_term
+            prec_vec[i] = prec_vec[i] + relative_precision[j]
+            i = i + 1
+        end 
+    end 
+    prec_vec[length(prec_vec)] = prec_vec[length(prec_vec)] + relative_precision[length(hodge_numbers)]
+
+    return prec_vec 
+end 
+
+"""
+    sign_fe(n, d, prec_vec, cp_coeffs)
+
+returns the sign of the functional equation. The sign is always 1 if the dimension of the hypersurface is odd. 
+
+INPUTS: 
+* "n" -- integer, dimension of the ambient projective space 
+* "d" -- integer, degree of characteristic polygon = dimension of cohomology group 
+* "prec_vec" -- list, prec_vec[i] = maximum p-adic valuation of the i-th coefficient of the Frobenius char poly? 
+* "cp_coeffs" -- list, cp_coeffs[i] = i-th coefficient of the Frobenius char poly, which a priori may be incorrect 
+"""
+function sign_fe(n, d, prec_vec, cp_coeffs)
+    sign = 1
+    dimension = n-1 # dimension of hypersurface = motivic weight = n-1 
+    if dimension % 2 == 0 
+        for i in 1:Int(floor(d/2))
+            # NOTE-TO-SELF: check if the symmetric index is d-i or d+1-i
+            p_power = p^(min(prec_vec[i], prec_vec[d-i]+((d-2*i)*dimension)//2))
+            if (mod(cp[i], p_power) != 0) && (mod(cp[d-i], p_pwer) != 0)
+                if 0 == mod(cp[i] + cp[d-i] * p^(((d-2*i)*dimension)//2), p_power)
+                    sign = -1
+                else
+                    sign = 1 
+                    @assert 0 == mod(cp[i] - cp[d-i] * p^(((d-2*i)*dimension)//2), p_power)
+                end 
+            end 
+        end 
+    end 
+
+    return sign 
+end 
+
+"""
+    apply_symmetry(n, d, p, prec_vec, cp_coeffs, mod, sign)
+
+Update the coefficients in the Frobenius characteristic polynomial using symmetry in the coefficients coming from 
+the Weil conjecture/Poincare duality 
+
+INPUTS: 
+* "n" -- integer, dimension of the ambient projective space 
+* "d" -- integer, degree of characteristic polygon = dimension of cohomology group 
+* "p" -- integer, prime number 
+* "prec_vec" -- list, prec_vec[i] = maximum p-adic valuation of the i-th coefficient of the Frobenius char poly? 
+* "cp_coeffs" -- list, cp_coeffs[i] = i-th coefficient of the Frobenius char poly, which a priori may be incorrect 
+* "mod" -- list, mod[i] = p^(prec_vec[i])
+* "sign" -- integer, sign of the functional equation
+"""
+function apply_symmetry(n, d, p, prec_vec, cp_coeffs, mod, sign)
+    dimension = n-1
+    for i in 1:(div(Pdeg, 2) + 1)
+        k = ((d-2*i) * dimension)//2 
+        if prec_vec[i] >= prec_vec[d-i] + k 
+            prec_vec[d-i] = prec_vec[i] - k 
+            mod[d-i] = p^(prec_vec[d-i])
+            cp_coeffs[d-i] = (sign * cp_coeffs[i])//p^k  # ?
+            cp_coeffs[d-i] = mod(cp_coeffs[d-i], mod[d-i])
+        else
+            prec_vec[i] = prec[d-i] + k 
+            mod[i] = p^(prec_vec[i])
+            cp_coeffs[i] = sign * cp_coeffs[d-i] * p^k
+            cp_coeffs[i] = mod(cp_coeffs[i], mod[i]) 
+        end 
+    end 
+
+    return cp_coeffs 
+end 

src/DeRham.jl

@@ -20,6 +20,9 @@ include("ControlledReduction.jl")
 include("Frobenius.jl")
 include("FinalReduction.jl")
 
+include("SlopesPolygon.jl")
+include("CharPolyFrob.jl")
+
 
 include("ZetaFunction.jl")