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<p>Let <i>p</i> be an odd prime number and let <i>K</i> be an imaginary quadratic field in which <i>p</i> is split. Let <i>f</i> be a modular form with good reduction at <i>p</i>. We study the variation of the Bloch–Kato Selmer groups and the Bloch–Kato–Shafarevich–Tate groups of <i>f</i> over the anticyclotomic <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textbf{Z}_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-extension <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(K_\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> of <i>K</i>. In particular, we show that under the generalized Heegner hypothesis, if the localization of the generalized Heegner cycle attached to <i>f</i> at one of the primes above <i>p</i> is primitive and certain local conditions hold, then the Pontryagin dual of the Selmer group of <i>f</i> over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(K_\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> is free over the Iwasawa algebra. Consequently, the Bloch–Kato–Shafarevich–Tate groups of <i>f</i> vanish. This generalizes earlier works of Matar and Matar–Nekovář on elliptic curves. Furthermore, our proof applies uniformly to the ordinary and non-ordinary settings.</p>

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On the structure of the Bloch–Kato Selmer groups of modular forms over anticyclotomic \(\textbf{Z}_p\)-towers

  • 期刊论文

  • 发表日期：

    2025年12月22日

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  • Antonio Lei,
  • Luca Mastella,
  • Luochen Zhao

摘要

Let p be an odd prime number and let K be an imaginary quadratic field in which p is split. Let f be a modular form with good reduction at p. We study the variation of the Bloch–Kato Selmer groups and the Bloch–Kato–Shafarevich–Tate groups of f over the anticyclotomic \(\textbf{Z}_p\) Z p -extension \(K_\infty \) K ∞ of K. In particular, we show that under the generalized Heegner hypothesis, if the localization of the generalized Heegner cycle attached to f at one of the primes above p is primitive and certain local conditions hold, then the Pontryagin dual of the Selmer group of f over \(K_\infty \) K ∞ is free over the Iwasawa algebra. Consequently, the Bloch–Kato–Shafarevich–Tate groups of f vanish. This generalizes earlier works of Matar and Matar–Nekovář on elliptic curves. Furthermore, our proof applies uniformly to the ordinary and non-ordinary settings.