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Report on the irreducibility of L-functions

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Number Theory, Analysis and Geometry

Abstract

In this paper, in honor of the memory of Serge Lang, we apply ideas of Chavdarov and work of Larsen to study the \(\mathbb{Q}\)-irreducibility, or lack thereof, of various orthogonal L-functions, especially L-functions of elliptic curves over function fields in one variable over finite fields. We also discuss two other approaches to these questions, based on work of Matthews, Vaserstein, and Weisfeller, and on work of Zalesskii-Serezkin.

Mathematics Subject Classification (2010): 11M38, 11M50, 14D10, 14D05

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Notes

  1. 1.

    The reason we expect this Weyl group is the fact [Weyl, (9.15) on p. 226] that in the compact orthogonal group \(O(2n + 2, \mathbb{R})\), the space of conjugacy classes of sign (here sign = determinant) − 1 is, with its “Hermann Weyl measure” of total mass one, isomorphic to the space of conjugacy classes in the compact symplectic group USp(2n), with its “Hermann Weyl measure” of total mass one.

  2. 2.

    Unlike the Larsen result or the Zalesskii–Serezkin result to be discussed below, this result [MVW] depends upon the classification of finite simple groups.

  3. 3.

    A third approach would be to appeal to the results of Hall [Ha].

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Fine Hall, Princeton, NJ, 08544-1000, USA

    Nicholas M. Katz

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  1. Nicholas M. Katz
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Correspondence to Nicholas M. Katz .

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Editors and Affiliations

  1. , Department of Mathematics, Columbia University, Broadway 2990, New York, 10027, New York, USA

    Dorian Goldfeld

  2. , Department of Mathematics, City College of New York, New York, 10031, New York, USA

    Jay Jorgenson

  3. , Department of Mathematics, Yale University, Hillhouse Ave. 10, New Haven, 06520, Connecticut, USA

    Peter Jones

  4. , Department of Mathematics, California Institute of Technology, E. California Blvd. 1200, Pasadena, 91125, California, USA

    Dinakar Ramakrishnan

  5. , Department of Mathematics, University of California at Berkeley, Evans Hall 925, Berkeley, 94720-3840, California, USA

    Kenneth Ribet

  6. , Department of Mathematics, Harvard University, Science Center/One Oxford Street, Cambridge, 02138, Massachusetts, USA

    John Tate

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Dedicated to the memory of Serge Lang

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Katz, N.M. (2012). Report on the irreducibility of L-functions. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_15

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