Trac #24108: quadratic_form_from_invariants()

This is algorithm 3.4.3 from Markus Kirschmer's "Definite quadratic and
hermitian forms with small class number"  It returns a global quadratic
form corresponding to a given set of local invariants.

URL: https://trac.sagemath.org/24108
Reported by: annahaensch
Ticket author(s): Simon Brandhorst
Reviewer(s): Travis Scrimshaw

src/sage/quadratic_forms/all.py

@@ -3,7 +3,7 @@
 
 from .ternary_qf import TernaryQF, find_all_ternary_qf_by_level_disc, find_a_ternary_qf_by_level_disc
 
-from .quadratic_form import QuadraticForm, DiagonalQuadraticForm
+from .quadratic_form import QuadraticForm, DiagonalQuadraticForm, quadratic_form_from_invariants
 
 from .random_quadraticform import random_quadraticform, random_quadraticform_with_conditions, random_ternaryqf, random_ternaryqf_with_conditions
 

src/sage/quadratic_forms/quadratic_form.py

@@ -5,6 +5,7 @@
 
 - Jon Hanke (2007-06-19)
 - Anna Haensch (2010-07-01): Formatting and ReSTification
+- Simon Brandhorst (2019-10-15): :meth:``quadratic_form_from_invariants``
 """
 
 #*****************************************************************************
@@ -28,7 +29,7 @@
 from sage.rings.ring import Ring
 from sage.misc.functional import denominator, is_even, is_field
 from sage.arith.all import GCD, LCM
-from sage.rings.all import Ideal
+from sage.rings.all import Ideal, QQ
 from sage.rings.ring import is_Ring, PrincipalIdealDomain
 from sage.structure.sage_object import SageObject
 from sage.structure.element import is_Vector
@@ -75,6 +76,103 @@ def is_QuadraticForm(Q):
     return isinstance(Q, QuadraticForm)
 
 
+def quadratic_form_from_invariants(F, rk, det, P, sminus):
+    r"""
+    Return a rational quadratic form with given invariants.
+
+    INPUT:
+
+    - ``F`` -- the base field; currently only ``QQ`` is allowed
+    - ``rk`` -- integer; the rank
+    - ``det`` -- rational; the determinant
+    - ``P`` -- a list of primes where Cassel's Hasse invariant
+      is negative
+    - ``sminus`` -- integer; the number of negative eigenvalues
+      of any Gram matrix
+
+    OUTPUT:
+
+    - a quadratic form with the specified invariants
+
+    Let `(a_1, \ldots, a_n)` be the gram marix of a regular quadratic space.
+    Then Cassel's Hasse invariant is defined as
+
+    .. MATH::
+
+        \prod_{i<j} (a_i,a_j),
+
+    where `(a_i,a_j)` denotes the Hilbert symbol.
+
+    ALGORITHM:
+
+    We follow [Kir2016]_.
+
+    EXAMPLES::
+
+        sage: P = [3,5]
+        sage: q = quadratic_form_from_invariants(QQ,2,-15,P,1)
+        sage: q
+        Quadratic form in 2 variables over Rational Field with coefficients:
+        [ 5 0 ]
+        [ * -3 ]
+        sage: all(q.hasse_invariant(p)==-1 for p in P)
+        True
+    """
+    from sage.arith.misc import hilbert_symbol
+    # normalize input
+    if F!=QQ:
+        raise NotImplementedError('base field must be QQ. If you want this over any field, implement weak approximation.')
+    P = [ZZ(p) for p in P]
+    rk = ZZ(rk)
+    d = QQ(det).squarefree_part()
+    sminus = ZZ(sminus)
+    # check if the invariants define a global quadratic form
+    if d.sign() != (-1)**sminus:
+        raise ValueError("invariants do not define a rational quadratic form")
+    if rk == 1 and len(P) != 0:
+        raise ValueError("invariants do not define a rational quadratic form")
+    if rk == 2:
+        for p in P:
+            if QQ(-d).is_padic_square(p):
+                raise ValueError("invariants do not define a rational quadratic form")
+    if sminus % 4 in (2, 3) and len(P) % 2 == 0:
+        raise ValueError("invariants do not define a rational quadratic form")
+    D = []
+    while rk >= 2:
+        if rk >= 4:
+            if sminus > 0:
+                a = ZZ(-1)
+            else:
+                a = ZZ(1)
+        elif rk == 3:
+            Pprime = [p for p in P if hilbert_symbol(-1, -d, p)==1]
+            Pprime += [p for p in (2*d).prime_divisors()
+                       if hilbert_symbol(-1, -d, p)==-1 and p not in P]
+            if sminus > 0:
+                a = ZZ(-1)
+            else:
+                a = ZZ(1)
+            for p in Pprime:
+                if d.valuation(p) % 2 == 0:
+                    a *= p
+            assert all((a*d).valuation(p)%2==1 for p in Pprime)
+        elif rk == 2:
+            S = P
+            if sminus == 2:
+                S += [-1]
+            a = QQ.hilbert_symbol_negative_at_S(S,-d)
+            a = ZZ(a)
+        P = [p for p in P if hilbert_symbol(a, -d, p) == 1]
+        P += [p for p in (2*a*d).prime_divisors()
+              if hilbert_symbol(a, -d, p)==-1 and p not in P]
+        sminus = max(0, sminus-1)
+        rk = rk - 1
+        d = a*d
+        D.append(a.squarefree_part())
+    d = d.squarefree_part()
+    D.append(d)
+    return DiagonalQuadraticForm(QQ,D)
+
 
 class QuadraticForm(SageObject):
     r"""