On the Bloch–Kato conjecture for elliptic modular forms of square-free level

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• 作者：Mahesh Agarwal (1)
  Jim Brown (2)
  
• 关键词：Bloch–Kato conjecture ; Congruences among automorphic forms ; Galois representations ; Saito–Kurokawa correspondence ; Siegel modular forms ; Special values of $$L$$ L ; functions ; Primary 11F33 ; 11F67 ; Secondary 11F46 ; 11F80
• 刊名：Mathematische Zeitschrift
• 出版年：2014
• 出版时间：April 2014
• 年：2014
• 卷：276
• 期：3-4
• 页码：889-924
• 全文大小：439 KB
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• 作者单位：Mahesh Agarwal (1)
  Jim Brown (2)
  
  1. Department of Mathematics and Statistics, University of Michigan-Dearborn, Dearborn, MI, 48128, USA
  2. Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA
  
• ISSN：1432-1823

文摘

Let $\kappa \ge 6$ be an even integer, $M$ an odd square-free integer, and $f \in S_{2\kappa -2}(\Gamma _0(M))$ a newform. We prove that under some reasonable assumptions that half of the $\lambda $ -part of the Bloch–Kato conjecture for the near central critical value $L(\kappa ,f)$ is true. We do this by bounding the $\ell $ -valuation of the order of the appropriate Bloch–Kato Selmer group below by the $\ell $ -valuation of algebraic part of $L(\kappa ,f)$ . We prove this by constructing a congruence between the Saito–Kurokawa lift of $f$ and a cuspidal Siegel modular form.