Let F be a newform of weight 2k on Γ0(N)\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _0(N)$$\end{document} with an odd integer N and a positive integer k, and ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} be a prime larger than or equal to 5 with (ℓ,N)=1\documentclass[12pt]{minimal}
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\begin{document}$$(\ell ,N)=1$$\end{document}. For each fundamental discriminant D, let χD\documentclass[12pt]{minimal}
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\begin{document}$$\chi _{D}$$\end{document} be a quadratic character associated with quadratic field Q(D)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Q}(\sqrt{D})$$\end{document}. Assume that for each D, the ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}-adic valuation of the algebraic part of L(F⊗χD,k)\documentclass[12pt]{minimal}
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\begin{document}$$L(F\otimes \chi _{D},k)$$\end{document} is non-negative. Let Wℓ+\documentclass[12pt]{minimal}
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\begin{document}$$W_{\ell }^{+}$$\end{document} (resp. Wℓ-\documentclass[12pt]{minimal}
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\begin{document}$$W_{\ell }^{-}$$\end{document}) be the set of positive (resp. negative) fundamental discriminants D with (D,N)=1\documentclass[12pt]{minimal}
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\begin{document}$$(D,N)=1$$\end{document} such that the ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}-adic valuation of the algebraic part of L(F⊗χD,k)\documentclass[12pt]{minimal}
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\begin{document}$$L(F\otimes \chi _{D},k)$$\end{document} is zero. We prove that for each sign ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon $$\end{document}, if Wℓϵ\documentclass[12pt]{minimal}
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\begin{document}$$W_{\ell }^{\epsilon }$$\end{document} is a non-empty finite set, then Wℓϵ⊂1,(-1)ℓ-12ℓ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} W_{\ell }^{\epsilon } \subset \left\{ 1, (-1)^{\frac{\ell -1}{2}}\ell \right\} .\end{aligned}$$\end{document}By this result, we prove that if ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon $$\end{document} is the sign of (-1)k\documentclass[12pt]{minimal}
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\begin{document}$$(-1)^k$$\end{document}, then k≥ℓ-1ork=ℓ-12.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}k\ge \ell -1 \text { or } k=\frac{\ell -1}{2}. \end{aligned}$$\end{document}These are applied to obtain a lower bound for #{D∈Wℓϵ:|D|≤X}\documentclass[12pt]{minimal}
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\begin{document}$$\#\{D\in W_{\ell }^{\epsilon } : |D|\le X \}$$\end{document} and the indivisibility of the order of the Shafarevich–Tate group of an elliptic curve over Q\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Q}$$\end{document}. To prove these results, first we refine Waldspurger’s formula on the Shimura correspondence for general odd levels N. Next we study mod ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} modular forms of half-integral weight with few non-vanishing coefficients. To do this, we use the filtration of mod ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} modular forms and mod ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} Galois representations.
