CONGRUENCES BETWEEN EXTREMAL MODULAR FORMS AND THETA SERIES OF SPECIAL TYPES MODULO POWERS OF 2 AND 3 CONGRUENCES BETWEEN EXTREMAL MODULAR FORMS AND THETA SERIES OF SPECIAL TYPES MODULO POWERS OF 2 AND 3

Access this Article

Search this Article

Author(s)

Abstract

Heninger, Rains and Sloane proved that the nth root of the theta series of any extremal even unimodular lattice in R^n has integer coefficients if n is of the form 2^i3^j5^k(i ≥ 3). Motivated by their discovery, we find the congruences between extremal modular forms and theta series of special types modulo powers of 2 and 3. This assertion enables us to prove that the 2nth root and the (3n/2)th root of the extremal modular form of weight n/2 have at least one non-integer coefficient.

Heninger, Rains and Sloane proved that the <I>n</I>th root of the theta series of any extremal even unimodular lattice in <B>R</B><SUP><I>n</I></SUP> has integer coefficients if n is of the form 2<SUP><I>i</I></SUP>3<SUP><I>j</I></SUP>5<SUP><I>k</I></SUP>(i ≥ 3). Motivated by their discovery, we find the congruences between extremal modular forms and theta series of special types modulo powers of 2 and 3. This assertion enables us to prove that the 2<I>n</I>th root and the (3<I>n</I>/2)th root of the extremal modular form of weight <I>n</I>/2 have at least one non-integer coefficient.

Journal

  • Kyushu Journal of Mathematics

    Kyushu Journal of Mathematics 63(1), 123-132, 2009

    Faculty of Mathematics, Kyushu University

Codes

  • NII Article ID (NAID)
    130000135274
  • NII NACSIS-CAT ID (NCID)
    AA10994346
  • Text Lang
    ENG
  • Article Type
    Departmental Bulletin Paper
  • ISSN
    1340-6116
  • Data Source
    IR  J-STAGE 
Page Top