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- W2093296136 abstract "For each prime $ell$, let $|cdot|_ell$ be an extension to $bar Q$ of the usual $ell$-adic absolute value on $Q$. Suppose $g(z) = sum_{n=0}^infty c(n)q^n in M_{k+half}(N)$ is an eigenform whose Fourier coefficients are algebraic integers. Under a mild condition, for all but finitely many primes $ell$ there are infinitely many square-free integers $m$ for which $|c(m)|_ell = 1$. Consequently we obtain indivisibility results for ``algebraic parts'' of central critical values of modular $L$-functions and class numbers of imaginary quadratic fields. These results partially answer a conjecture of Kolyvagin regarding Tate-Shafarevich groups of modular elliptic curves. Similar results were obtained earlier by Jochnowitz by a completely different method. Our method uses standard facts about Galois representations attached to modular forms, and pleasantly uncovers surprising Kronecker-style congruences for $L$-function values. For example if $Delta(z)$ is Ramanujan's cusp form and $g(z)=sum_{n=1}^{infty}c(n)q^n$ is the cusp form for which $$L(Delta_D,6)=fracwithdelims(){pi}{D}^6frac{sqrt{D}}{5!}frac{ } { }cdot c(D)^2,$$ for fundamental discriminants $D>0,$ then for $Ngeq 1$ $$sum_{k=-infty}^infty c(N-k^2) equiv half sum_{d|N}(chi_{-1}(d)+chi_{-1}(N/d))d^6 pmod {61}. tag{0}$$" @default.
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- W2093296136 date "1996-11-01" @default.
- W2093296136 modified "2023-09-27" @default.
- W2093296136 title "Fourier coefficients of half-integral weight modular forms modulo ell" @default.
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