Adjustments for the 2-adic lseries of elliptic curves

[chriswuthrich]

This is a minor bug that I am sure to be the first one to notice. Currently one cannot evaluate the (unique) Teichmuller twist of the 2-adic L-series of an elliptic curve. Furthermore the odd powers of the Teichmuller twists are incorrectly normalised. This is harmless except for p=2.

CC: @loefflerd

Component: elliptic curves

Keywords: p-adic L-series

Author: Chris Wuthrich

Branch: 7b7f6ad

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/32258

[chriswuthrich]

comment:1

Expect a correction soon.

[chriswuthrich]

Commit: f073844

[chriswuthrich]

comment:2

But I couldn't do any testing yet or adjusting doctests.

---

New commits:

f073844

adjust 2-adic lseries for elliptic curves

[chriswuthrich]

Branch: u/wuthrich/ticket_32258

[chriswuthrich]

Author: Chris Wuthrich

[chriswuthrich]

comment:3

I will record here, why the change in the normalisation of the odd powers of the Teichmüller twists are now correct.

Suppose E has good ordinary reduction at p. Let L_p(E) be the p-adic L-series of E as sage returns it. If p=2 or p%4 = 1, then the p-adic L-series twisted by the (p-1)/2-th power of the Teichmüller corresponds to the twist by the quadratic character chi by -1 or -p respectively. Let L_p(E,chi) to be this p-adic L-series returned by taking eta=1 or eta=(p-1)/2 respectively. The constant term of L_p(E,chi) is equal to 1/alpha^u * L(E,chi,1) * Gausssum(chi)/ Omega_minus with u=2 if p=2 and u=1 if p>2. The question is now why is Omega_minus equal to the smallest positive multiple of i that is in the period lattice of E.

The main conjecture says that L_p(E) and the twisted L_p(E,chi) are the characteristic series of Selmer groups. Greenberg has calculated the constant term even in the case p=2. Suppose E has rank 0 over K = Q(i) or K = Q(sqrt(-p) respectively. The main conjecture now predict L_p(E)(0) * L_p(E,chi)(0) and (1-1/alpha)^2 * 1/alpha^u * prod c_v(E/K) * |Sha(E/K)|/ |E(K)|^2, where the product runs over all FINITE Tamagawa numbers, have the same p-adic valuation. The p-adic BSD of E/K predicts the equality of these two p-adic numbers.

The following function should therefore return a p-adic approximation of a power of 4 if the conditions above are satisfied.

def ch(E,p):
    """
    check curve E at p,
    should return a power of 4.

    EXAMPLE::

        sage: ch(EllipticCurve("443c1"), 13)
        1 + O(13^5)
        sage: ch(EllipticCurve("443c1"), 2)
        2^2 + O(2^8)

    """
    if p == 2:
        u = 2
        D = -1
        po = 1
        k = 8
    else:
        u = 1
        D = -p
        po = (p-1)//2
        k = 4 if p<6 else 3

    lp = E.padic_lseries(p,implementation="num")
    l0 = lp.series(k, prec=3)
    l1 = lp.series(k, eta = po, prec=3)
    ct = l0.list()[0] * l1.list()[0]

    al = lp.alpha()
    eps = (1-1/al)**2 * 1/al**u

    K = QuadraticField(D)
    EK = E.base_extend(K)
    Ed = E.quadratic_twist(D)
    tam = EK.tamagawa_product()
    sh = E.sha().an() * Ed.sha().an() # that is not SHA/K because of 2-torsion.
    tor = EK.torsion_order()
    return ct/eps/tam/sh*tor**2

... and I have checked it on quite a number of curves with both rectamgular and non-rectangular period lattice.

[sagetrac-git]

Changed commit from f073844 to 5620235

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

5620235

trac 32258: add doctest

[fchapoton]

comment:7

there is a typo in "charachters"

[sagetrac-git]

Changed commit from 5620235 to 7b7f6ad

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

b859fd6	Merge branch 'develop' of git://github.com/sagemath/sage into twoadiclseries
7b7f6ad	trac 32258: typos

[fchapoton]

Reviewer: Frédéric Chapoton

[fchapoton]

comment:9

I will assume that the math is correct => positive review.

[vbraun]

Changed branch from u/wuthrich/ticket_32258 to 7b7f6ad

[fchapoton]

Changed commit from 7b7f6ad to none