Premium
Linear dependence of certain L ‐values of half‐integral weight modular forms
Author(s) -
Katsurada Hidenori,
Mizuno Yoshinori
Publication year - 2012
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdr057
Subject(s) - mathematics , modular form , lift (data mining) , cusp form , holomorphic function , integer (computer science) , pure mathematics , convolution (computer science) , combinatorics , machine learning , computer science , artificial neural network , data mining , programming language
Let h be a cusp form of half‐integral weight, andε 2 / 3 ( χ )be a certain (not necessarily holomorphic) modular form of weight3 2associated with a Dirichlet character χ. We then prove linear dependence of the valuesR ( l , h , ε 2 / 3 ( χ ) )of the Rankin–Selberg convolution products of h andε 2 / 3 ( χ )when we fix an integer l and vary χ. A main idea for the proof is to express such a convolution product as the twisted Koecher–Maass series L ( s, M ( h ), χ) for the Maass lift M ( h ) of h , whose values at integers were investigated by Choie and Kohnen. Moreover, by using such an expression, we obtain an algebraicity result for the values of L ( s, M ( h ), χ) at half‐integers, which was not considered by Choie and Kohnen.