CoCalc -- unfiled.py

Harvard Workshop: Distribution of modular symbols and L-values
code / alex / psage / psage / modform / rational / unfiled.py
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#################################################################################
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#
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# (c) Copyright 2010 William Stein
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#
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#  This file is part of PSAGE
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#
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#  PSAGE is free software: you can redistribute it and/or modify
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#  it under the terms of the GNU General Public License as published by
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#  the Free Software Foundation, either version 3 of the License, or
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#  (at your option) any later version.
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# 
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#  PSAGE is distributed in the hope that it will be useful,
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#  but WITHOUT ANY WARRANTY; without even the implied warranty of
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#  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
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#  GNU General Public License for more details.
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# 
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#  You should have received a copy of the GNU General Public License
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#  along with this program.  If not, see <http://www.gnu.org/licenses/>.
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#
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#################################################################################
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"""
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Miscellaneous code related to rational modular forms, until I find a
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better place to put it.
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WARNING: Depend on this code at your own risk!  It will get moved to
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another module at some point, probably.
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"""
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def modular_symbols_from_curve(C, N, num_factors=3):
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    """
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    Find the modular symbols spaces that shoudl correspond to the
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    Jacobian of the given hyperelliptic curve, up to the number of
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    factors we consider.
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    INPUT:
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        - C -- a hyperelliptic curve over QQ
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        - N -- a positive integer
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        - num_factors -- number of Euler factors to verify match up; this is
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          important, because if, e.g., there is only one factor of degree g(C), we
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          don't want to just immediately conclude that Jac(C) = A_f. 
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    OUTPUT:
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        - list of all sign 1 simple modular symbols factor of level N
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          that correspond to a simple modular abelian A_f
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          that is isogenous to Jac(C).  
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    EXAMPLES::
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        sage: from psage.modform.rational.unfiled import modular_symbols_from_curve
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        sage: R.<x> = ZZ[]
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        sage: f = x^7+4*x^6+5*x^5+x^4-3*x^3-2*x^2+1
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        sage: C1 = HyperellipticCurve(f)
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        sage: modular_symbols_from_curve(C1, 284)
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        [Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 39 for Gamma_0(284) of weight 2 with sign 1 over Rational Field]
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        sage: f = x^7-7*x^5-11*x^4+5*x^3+18*x^2+4*x-11
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        sage: C2 = HyperellipticCurve(f)
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        sage: modular_symbols_from_curve(C2, 284)
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        [Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 39 for Gamma_0(284) of weight 2 with sign 1 over Rational Field]
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    """
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    # We will use the Eichler-Shimura relation and David Harvey's
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    # p-adic point counting hypellfrob.  Harvey's code has the
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    # constraint:   p > (2*g + 1)*(2*prec - 1).
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    # So, find the smallest p not dividing N that satisfies the
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    # above constraint, for our given choice of prec.
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    f, f2 = C.hyperelliptic_polynomials()
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    if f2 != 0:
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        raise NotImplementedError, "curve must be of the form y^2 = f(x)"
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    if f.degree() % 2 == 0:
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        raise NotImplementedError, "curve must be of the form y^2 = f(x) with f(x) odd"
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    prec = 1
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    g = C.genus()
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    B = (2*g + 1)*(2*prec - 1)
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    from sage.rings.all import next_prime
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    p = B
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    # We use that if F(X) is the characteristic polynomial of the
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    # Hecke operator T_p, then X^g*F(X+p/X) is the characteristic
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    # polynomial of Frob_p, because T_p = Frob_p + p/Frob_p, according
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    # to Eichler-Shimura.  Use this to narrow down the factors. 
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    from sage.all import ModularSymbols, Integers, get_verbose
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    D = ModularSymbols(N,sign=1).cuspidal_subspace().new_subspace().decomposition()
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    D = [A for A in D if A.dimension() == g]
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    from sage.schemes.hyperelliptic_curves.hypellfrob import hypellfrob
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    while num_factors > 0:
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        p = next_prime(p)
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        while N % p == 0: p = next_prime(p)
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        R = Integers(p**prec)['X']
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        X = R.gen()
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        D2 = []
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        # Compute the charpoly of Frobenius using hypellfrob
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        M = hypellfrob(p, 1, f)
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        H = R(M.charpoly())
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        for A in D:
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            F = R(A.hecke_polynomial(p))
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            # Compute charpoly of Frobenius from F(X)
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            G = R(F.parent()(X**g * F(X + p/X)))
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            if get_verbose(): print (p, G, H)
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            if G == H:
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                D2.append(A)
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        D = D2
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        num_factors -= 1
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    return D
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