4,082 research outputs found

    Twisted Hecke L-values and period polynomials

    Get PDF
    Let f1,...,fdf_1,...,f_d be an orthogonal basis for the space of cusp forms of even weight 2k2k on Ξ“0(N)\Gamma_0(N). Let L(fi,s)L(f_i,s) and L(fi,Ο‡,s)L(f_i,\chi,s) denote the LL-function of fif_i and its twist by a Dirichlet character Ο‡\chi, respectively. In this note, we obtain a ``trace formula'' for the values L(fi,Ο‡,m)L(fi,n)β€ΎL(f_i,\chi,m)\overline{L(f_i,n)} at integers mm and nn with 0<m,n<2k0<m,n<2k and proper parity. In the case N=1 or N=2, the formula gives us a convenient way to evaluate precisly the value of the ratio L(f,Ο‡,m)/L(f,n)L(f,\chi,m)/L(f,n) for a Hecke eigenform ff.Comment: 20 page

    Differential equations satisfied by modular forms and K3 surfaces

    Full text link
    We study differential equations satisfied by modular forms associated to Ξ“1Γ—Ξ“2\Gamma_1\times\Gamma_2, where Ξ“i(i=1,2)\Gamma_i (i=1,2) are genus zero subgroups of SL2(R)SL_2(\mathbf R) commensurable with SL2(Z)SL_2(\mathbf Z), e.g., Ξ“0(N)\Gamma_0(N) or Ξ“0(N)βˆ—\Gamma_0(N)^*. In some examples, these differential equations are realized as the Picard--Fuch differential equations of families of K3 surfaces with large Picard numbers, e.g., 19,18,17,1619, 18, 17, 16. Our method rediscovers some of the Lian--Yau examples of ``modular relations'' involving power series solutions to the second and the third order differential equations of Fuchsian type in [14, 15].Comment: Some revisions are incorporated, in particular, replaced the terminology ''bi-modular'' by ''modular'
    • …
    corecore