From 6fe120b690d845fd8567181ba0059da772e54a21 Mon Sep 17 00:00:00 2001
From: jgarnek <jgarnek@amu.edu.pl>
Date: Mon, 10 Jun 2024 19:08:42 +0000
Subject: [PATCH] group and template

---
 as_covers/group.sage    | 99 +++++++++++++++++++++++++++++++++++++++++
 as_covers/template.sage | 37 +++++++++++++++
 2 files changed, 136 insertions(+)
 create mode 100644 as_covers/group.sage
 create mode 100644 as_covers/template.sage

diff --git a/as_covers/group.sage b/as_covers/group.sage
new file mode 100644
index 0000000..6b6f9ff
--- /dev/null
+++ b/as_covers/group.sage
@@ -0,0 +1,99 @@
+class group:
+    def __init__(self, name, elts, one, mult, inv, gens, as_gens):
+        self.name = name
+        self.elts = elts
+        self.one = one
+        self.mult = mult
+        self.inv = inv
+        self.order = len(self.elts)
+        self.gens = gens
+        self.as_gens = as_gens
+        
+    def __repr__(self):
+        return self.name
+    
+    def elt(self, a_tuple):
+        return group_elt(a_tuple, self)
+    
+    def ONE(self):
+        return self.elt(self.one)
+    
+    def GENS(self):
+        return [self.elt(aa) for aa in self.gens]
+    
+class group_elt:
+    def __init__(self, as_tuple, group):
+        self.group = group
+        self.as_tuple = as_tuple
+        self.as_gens = self.group.as_gens(as_tuple)
+        
+    def __repr__(self):
+        return str(self.as_tuple)
+    
+    def __mul__(self, other):
+        result_as_tuple = self.group.mult(self.as_tuple, other.as_tuple)
+        return group_elt(result_as_tuple, self.group)
+    
+    def __neg__(self):
+        result_as_tuple = self.group.inv(self.as_tuple)
+        return group_elt(result_as_tuple, self.group)
+    
+    def __pow__(self, m):
+        if m == 0:
+            return self.group.ONE()
+        if m == 1:
+            return self
+        if m == 2:
+            return self*self
+        if m < 0:
+            return -(self^(-m))
+        if m%2 == 1:
+            return (self^(m//2))^2*self
+        return (self^(m//2))^2
+    
+    def __eq__(self, other):
+        return self.as_tuple == other.as_tuple
+    
+    
+def cyclic_gp(p, n):
+    name = "cyclic group of order " + str(p) + "^" + str(n)
+    elts = [i for i in range(p^n)]
+    one = 0
+    mult = lambda i1, i2: (i1 + i2) % (p ** n)
+    inv = lambda i: (-i) % (p ** n)
+    gens = [1]
+    as_gens = lambda i : [[0, i]]
+    gp = group(name, elts, one, mult, inv, gens, as_gens)
+    return gp
+
+def elementary_gp(p, n):
+    name = "(Z/" + str(p) + ")" + "^" + str(n)
+    pr = [list(GF(p)) for _ in range(n)]
+    from itertools import product
+    elts = []
+    for a in product(*pr):
+        elts += [a]
+    one = elts[0]
+    mult = lambda i1, i2: [(i1[j] + i2[j]) % p for j in range(n)]
+    inv = lambda i: [(-i[j]) % p for j in range(n)]
+    gens = []
+    for i in range(n):
+        e = n*[0]
+        e[i] = 1
+        gens += [tuple(e)]
+    as_gens = lambda i : [[j, i[j]] for j in range(n)]
+    gp = group(name, elts, one, mult, inv, gens, as_gens)
+    return gp
+
+
+
+def heisenberg(p):
+    name = "Heisenberg group E(" + str(p) + "^3)"
+    elts = [(i, j, k) for i in range(p) for j in range(p) for k in range(p)]
+    one = 0
+    mult = lambda elt1, elt2 : ((elt1[0] + elt2[0])%p, (elt1[1] + elt2[1])%p, (-elt1[0]*elt2[1] + elt1[2] + elt2[2])%p)
+    inv = lambda elt : (p-elt[0], p-elt[1], (p - elt[2] - (p-elt[0])*(p-elt[1]))%p)
+    gens = [(1, 0, 0), (0, 1, 0), (0, 0, 1)]
+    as_gens = lambda elt : [[0, elt[0]], [1, elt[1]], [2, elt[2]]]
+    gp = group(name, elts, one, mult, inv, gens, as_gens)
+    return gp
\ No newline at end of file
diff --git a/as_covers/template.sage b/as_covers/template.sage
new file mode 100644
index 0000000..2facb38
--- /dev/null
+++ b/as_covers/template.sage
@@ -0,0 +1,37 @@
+class template:
+    '''Template of a p-group cover'''
+    def __init__(self, height, field, group, fcts, gp_action):
+        self.height = height
+        self.group = group
+        self.fcts = fcts #RHSs of the Artin-Schreier equations
+        self.gp_action = gp_action #action of the generators of the group on z[i]'s
+        self.field = field
+        n = height
+        variable_names = ''
+        for i in range(n):
+            variable_names += 'z'+str(i)+','
+        for i in range(n):
+            variable_names += 'f'+str(i)
+            if i!=n-1:
+                variable_names += ','
+        R = PolynomialRing(field, 2*n, variable_names)
+        z = R.gens()[:n]
+        f = R.gens()[n:]
+
+def elementary_template(p, n):
+    group = elementary_gp(p, n)
+    field = GF(p)
+    variable_names = ''
+    for i in range(n):
+        variable_names += 'z'+str(i)+','
+    for i in range(n):
+        variable_names += 'f'+str(i)
+        if i!=n-1:
+            variable_names += ','
+    R = PolynomialRing(field, 2*n, variable_names)
+    z = R.gens()[:n]
+    f = R.gens()[n:]
+    height = n
+    fcts = [f[i] for i in range(n)]
+    gp_action = [[z[j] + (i == j) for j in range(n)] for i in range(n)]
+    return template(height, field, group, fcts, gp_action)
\ No newline at end of file