August 9, 2007

  • Amalfi Conjecture:

    "Serious numbers  

    will always be heard."


    -- Paul Simon

    (See St. Luke's Day, 2005.)  


    Bulletin of the American Mathematical Society    ,

    Volume 31, Number 1, July 1994, Pages 1-14

    Selberg's Conjectures
    and Artin L-Functions

    (pdf)

    M. Ram Murty

    Introduction

    In its comprehensive form, an identity between an automorphic
    L-function and
    a "motivic" L-function is called a reciprocity law. The celebrated
    Artin reciprocity
    law is perhaps the fundamental example. The conjecture of
    Shimura-Taniyama that
    every elliptic curve over Q is "modular" is certainly the most
    intriguing reciprocity
    conjecture of our time. The "Himalayan peaks" that hold the secrets of
    these
    nonabelian reciprocity laws challenge humanity, and, with the visionary
    Langlands
    program    , we have mapped out before us one means of ascent to those
    lofty peaks.
    The recent work of Wiles suggests that an important case (the
    semistable case)
    of the Shimura-Taniyama conjecture is on the horizon and perhaps this
    is another
    means of ascent. In either case, a long journey is predicted.... At the
    1989 Amalfi meeting, Selberg [S] announced a series of
    conjectures which
    looks like another approach to the summit. Alas, neither path seems the
    easier
    climb....


    [S]     A. Selberg, Old and new
          conjectures and results
          about a class of Dirichlet series,
          Collected Papers, Volume II,
          Springer-Verlag, 1991, pp. 47-63.

    Zentralblatt MATH Database on the above Selberg paper:

    "These are notes of lectures presented at the Amalfi Conference on Number Theory, 1989.... There are various stimulating conjectures
    (which are related to several other conjectures like the Sato-Tate
    conjecture, Langlands conjectures, Riemann conjecture...).... Concluding
    remark of the author: 'A more complete account with proofs is under preparation and will in time appear elsewhere.'"

    Related material: Previous entry.