#32258 closed defect (fixed)

Adjustments for the 2-adic lseries of elliptic curves

Reported by: wuthrich Owned by:
Priority: minor Milestone: sage-9.5
Component: elliptic curves Keywords: elliptic curves, p-adic L-series
Cc: loeffler Merged in:
Authors: Chris Wuthrich Reviewers: Frédéric Chapoton
Report Upstream: N/A Work issues:
Branch: 7b7f6ad (Commits, GitHub, GitLab) Commit: 7b7f6ad24620b744d4decd6dcf648bd479762948
Dependencies: Stopgaps:

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Description

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.

Change History (10)

comment:1 Changed 13 months ago by wuthrich

Expect a correction soon.

comment:2 Changed 13 months ago by wuthrich

  • Authors set to Chris Wuthrich
  • Branch set to u/wuthrich/ticket_32258
  • Commit set to f073844d8b3878b679d7858c03da999564dfca7a

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


New commits:

f073844adjust 2-adic lseries for elliptic curves

comment:3 Changed 13 months ago by wuthrich

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.

comment:4 Changed 13 months ago by git

  • Commit changed from f073844d8b3878b679d7858c03da999564dfca7a to 56202355f5df69bd015014ade3a7a25732138240

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

5620235trac 32258: add doctest

comment:5 Changed 13 months ago by wuthrich

  • Cc loeffler added
  • Status changed from new to needs_review

comment:6 Changed 12 months ago by mkoeppe

  • Milestone changed from sage-9.4 to sage-9.5

comment:7 Changed 11 months ago by chapoton

there is a typo in "charachters"

comment:8 Changed 11 months ago by git

  • Commit changed from 56202355f5df69bd015014ade3a7a25732138240 to 7b7f6ad24620b744d4decd6dcf648bd479762948

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

b859fd6Merge branch 'develop' of git://github.com/sagemath/sage into twoadiclseries
7b7f6adtrac 32258: typos

comment:9 Changed 11 months ago by chapoton

  • Reviewers set to Frédéric Chapoton
  • Status changed from needs_review to positive_review

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

comment:10 Changed 10 months ago by vbraun

  • Branch changed from u/wuthrich/ticket_32258 to 7b7f6ad24620b744d4decd6dcf648bd479762948
  • Resolution set to fixed
  • Status changed from positive_review to closed
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