Nonvanishing modulo \(l\) of Fourier coefficients of half-integral weight modular forms
In an earlier paper [Abh. Math. Sem. Univ. Hamb. 68, 163-168 (1998; Zbl 0954.11016)] the author proved results on the nonvanishing of Fourier coefficients of modular forms of half-integral weight. Now he proves stronger results on the indivisibility by primes for these coefficients. This is related to recent work by \textit{K. Ono} and \textit{C. Skinner} [Ann. Math. (2) 147, 453-470 (1998; Zbl 0907.11017)] and by \textit{J. Jimenez Urroz} and \textit{K. Ono} [Math. Res. Lett. 7, 205-212 (2000; Zbl 0958.11031)], but the author does not need the concept of ``good'' modular form which was used by Ono and his co-authors. As tools the author uses various operators on modular forms and the so-called \(q\)-expansion principle of arithmetic algebraic geometry. It states that if \(f\) is a modular form of weight \(k+ {1\over 2}\) on \(\Gamma_1(N)\) whose Fourier coefficients \(a(n)\) are algebraic integers, and if \(l\) is a prime not dividing \(N\), then \(f\mid W_N\) has algebraic Fourier coefficients and satisfies \(v_l(f\mid W_N) =v_l(f)\). Here, \(W_N\) is the Fricke involution, \(v_l\) is an extension of the \(l\)-adic valuation of \(\mathbb{Q}\) on the algebraic closure of \(\mathbb{Q}\), and \(v_l(f)= \inf\{v_l (a(n)): n\geq 0\}\). Now let \(f\) be a modular form of weight \(k+{1\over 2}\) on \(\Gamma_0 (N)\) with Nebentypus character \(\chi\) whose Fourier coefficients are algebraic integers and which is a common eigenform of all Hecke operators \(T(q^2)\) for primes \(q\). One of the author's main results states: If distinct primes \(p_1, \dots,p_r\) not dividing \(N\) and values \(\varepsilon_1, \dots\varepsilon_r \in\{-1,0,1\}\) are given, then for all but finitely many primes \(l\) (depending on \(N,p_1,\dots, p_r\) and the eigenvalues of \(f)\) with \(v_l(f)=0\), there exist infinitely many square-free integers \(d\) such that \(v_l(a(d))=0\) and \(({d\over p_j})= \varepsilon_j\) for \(j=1,\dots, r\). The author gives interesting applications on critical values of twisted \(L\)-series attached to newforms of even weight \(2k\), on the indivisibility of special values of Dirichlet \(L\)-series, and the following result on class numbers of imaginary-quadratic fields: If distinct odd primes \(p_1,\dots,p_r\) and values \(\varepsilon_1, \dots, \varepsilon_r\in \{-1,0,1\}\) are given, then for every prime \(l>5\) not dividing \(\prod_j p_j(p^2 j-1)\), there are infinitely many fundamental discriminants \(D<0\) for which the class number \(h(D)\) is not divisible by \(l\) and \(({D\over p_j})= \varepsilon_j\) for \(j=1,\dots,r\). This generalizes results due to \textit{K. Horie} [Math. Ann. 288, 605-612 (1990; Zbl 0692.12004)].
\(q\)-expansion principle of arithmetic algebraic geometry
indivisibility of special values of Dirichlet \(L\)-series
class numbers of imaginary-quadratic fields
scientific article; zbMATH DE number 1425231
scientific article