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Big Heegner point Kolyvagin system for a family of modular forms

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Abstract

The principal goal of this paper is to develop Kolyvagin’s descent to apply with the big Heegner point Euler system constructed by Howard  for the big Galois representation \(\mathbb T \) attached to a Hida family \(\mathbb F \) of elliptic modular forms. In order to achieve this, we interpolate and control the Tamagawa factors attached to each member of the family \(\mathbb F \) at bad primes, which should be of independent interest. Using this, we then work out the Kolyvagin descent on the big Heegner point Euler system so as to obtain a big Kolyvagin system that interpolates the collection of Kolyvagin systems obtained by Fouquet for each member of the family individually. This construction has standard applications to Iwasawa theory, which we record at the end.

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Notes

  1. In the sense of Nekovář [21, §12.7.10]. The twisted eigenform we have in mind is denoted by \(g_\mathcal{P }\) in loc.cit.

  2. Attentive reader will notice that Mazur and Rubin never treat Heegner points. It was Howard [16] who was the first to study the Heegner points from the perspective offered by the work of Mazur and Rubin.

  3. In the sense we have explained at the start of Sect. 3.1.

  4. The attentive reader will notice that the extra factor \(\alpha \) is not explicit in the statement of Fouquet [9, Theorem B(iii)], however that Fouquet’s element \(z_\infty \) differ from the \(\mathfrak z _{\infty }\) defined below by a factor of \(\alpha \).

  5. Note that the extra factor \(\alpha \) in Fouquet’s[9] arguments is needed to obtain Kolyvagin systems for each specialization of \(\mathbb T \). Once one obtains those (in our case, they descend from our big Heegner point Kolyvagin system), the arguments of Sect. 6 in loc.cit. carry out verbatim.

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Acknowledgments

The author wishes to thank Olivier Fouquet, Barry Mazur and Jan Nekovář, discussions with whom led the author to this work. He also thanks the anonymous referee for his comments and suggestions. The author acknowledges a Marie Curie Grant EU-FP7 230668 as well as partial support from TÜBİTAK and FONDECYT in the duration of this project.

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  1. Koç University, Mathematics, Rumeli Feneri Yolu, Sariyer, 34450 , Istanbul, Turkey

    Kâzım Büyükboduk

  2. PUC Facultad de Matemáticas, Campus San Joaquín Avenida Vicuña Mackenna, 4860 , Santiago, Chile

    Kâzım Büyükboduk

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Büyükboduk, K. Big Heegner point Kolyvagin system for a family of modular forms. Sel. Math. New Ser. 20, 787–815 (2014). https://doi.org/10.1007/s00029-013-0136-4

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