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Codes from Modular Curves

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Selected Unsolved Problems in Coding Theory

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

One of the most interesting class of curves, from the perspective of arithmetical algebraic geometry, are the so-called modular curves. Some of the most remarkable applications of algebraic geometry to coding theory arise from these modular curves. It turns out these algebraic-geometric codes (“AG codes”) constructed from modular curves can have parameters which beat the Gilbert–Varshamov lower bound if the ground field is sufficiently large.

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Notes

  1. 1.

    The expository paper [JS] discussed this in more detail from the computational perspective.

  2. 2.

    For some introductions to this highly technical work of Langlands and Kottwitz, the reader is referred to Labesse [Lab], Clozel [Cl], and Casselman [Cas2].

  3. 3.

    The space \(\mathbb{H}=\{ z \in\mathbb{C}\;|\; \mathrm{Im}\,(z) > 0\}\) is also called the Poincaré upper half plane.

  4. 4.

    This result was essentially first proved by Igusa [Ig] (from the classical perspective). See also [TV], Theorem 4.1.48, and [Cas1] for an interesting discussion of what happens at the “bad primes,” and Deligne’s paper in the same volume as [Cas1].

  5. 5.

    Type optional_packages() for the name of the latest version of this database. This loads both ClassicalModularPolynomialDatabase and AtkinModularPolynomialDatabase.

  6. 6.

    In fact, if we write \(f(z)=\,\sum _{n=1}^{\infty}a_{n}q^{n}\), then

    $$\zeta_C(s)=\bigl(1-p^{-s}\bigr)^{-1}\prod_{p\not= 11}\bigl(1-a_pp^{-s}+p^{1-2s}\bigr)^{-1}$$

    is the global Hasse–Weil zeta function of the elliptic curve C of conductor 11 with Weierstrass model y 2+y=x 3−x 2 [Gel] (p. 252).

  7. 7.

    The genus formulas for X 0(N) given in [Shim] and [Kn] both apparently contain a (typographical) error. The problem is in the μ 2 term, which should contain a Legendre symbol \(({\frac{-4}{n}})\) instead of \(({\frac{-1}{n}})\). See, for example, [Ei] for a correct generalization.

  8. 8.

    In other words, C has length 56, dimension 22 over \({\mathbb{F}}\), and minimum distance 32.

  9. 9.

    The conductor is defined in Ogg [O1], but see also [Gel], Sect. I.2, or [Kn], p. 390.

  10. 10.

    Recall Singleton’s bound: n≥d+k−1.

  11. 11.

    When p=3 it is a model of a modular curve of level 32 (see Table 6.1). When p=7 this example arises in the reduction of X(7) in characteristic 7 [E2].

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  1. Mathematics Department, US Naval Academy, Chauvenet Hall Holloway Road 572C, Annapolis, Maryland, 21402, USA

    David Joyner

  2. Department of Mathematics, University of Louisville, Natural Sciences Building 328, Louisville, KY, 40292, USA

    Jon-Lark Kim

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Joyner, D., Kim, JL. (2011). Codes from Modular Curves. In: Selected Unsolved Problems in Coding Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8256-9_6

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