Abstract
Fixing a nontrivial automorphism of a number field K, we associate to ideals in K an invariant (with values in {0,±1}) which we call the spin and for which the associated L-function does not possess Euler products. We are nevertheless able, using the techniques of bilinear forms, to handle spin value distribution over primes, obtaining stronger results than the analogous ones which follow from the technology of L-functions in its current state. The initial application of our theorem is to the arithmetic statistics of Selmer groups of elliptic curves.
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Notes
For Gaussian primes, the name “spin” was used in [6], but for a symbol which is only superficially reminiscent of our \(\operatorname {spin}(\mathfrak{p})\) for prime ideals. Writing π=r+is∈ℤ[i] uniquely with r,s>0,r odd, the spin of \(p=\pi\bar{\pi}\) was defined to be the (usual) Jacobi symbol σ p =(s/r)=±1.
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Acknowledgements
This work was initiated and a significant part was accomplished during the time the authors enjoyed participating, with MSRI financial support, in the program on “Arithmetic Statistics”, held at MSRI Berkeley during January–May, 2011. The second and third-named authors also received support from the Clay Mathematics Institute. Research of J.F. is supported by NSERC grant A5123, that of H.I. by NSF Grant DMS-1101574, that of B.M. by NSF Grant DMS-0968831 and that of K.R. by NSF Grant DMS-1065904. We would also like to thank the referee for a very thorough reading of the paper.
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© 2012 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
An erratum to this article is available at http://dx.doi.org/10.1007/s00222-015-0613-9.
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Friedlander, J.B., Iwaniec, H., Mazur, B. et al. The spin of prime ideals. Invent. math. 193, 697–749 (2013). https://doi.org/10.1007/s00222-012-0438-8
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DOI: https://doi.org/10.1007/s00222-012-0438-8