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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Michigan-Dearborn, Dearborn, MI, 48128, USA

    Mahesh Agarwal

  2. Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA

    Jim Brown

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  1. Mahesh Agarwal
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  2. Jim Brown
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Correspondence to Mahesh Agarwal.

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The second author was partially supported by the National Security Agency under Grant Number H98230-11-1-0137. The United States Government is authorized to reproduce and distribute reprints not-withstanding any copyright notation herein.

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Agarwal, M., Brown, J. On the Bloch–Kato conjecture for elliptic modular forms of square-free level. Math. Z. 276, 889–924 (2014). https://doi.org/10.1007/s00209-013-1226-x

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