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Shafarevich-Tate Groups of Nonsquare Order

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Modular Curves and Abelian Varieties

Part of the book series: Progress in Mathematics ((PM,volume 224))

Abstract

Let A denote an abelian variety over ℚ. We give the first known examples in which #Ш(A/ℚ) is neither a square nor twice a square. For example, let E be the elliptic curve y2 + y = x3 — x of conductor 37. We prove that for every odd prime p < 25000 (with p 37), there is a twist A of E x • • • x E (p —1 copies) such that #Ш(A/ℚ) = pn2 for some integer n. We prove this by showing under certain hypothesis on E and p that there is an exact sequence

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Authors and Affiliations

  1. 515 Science Center Department of Mathematics, Harvard University, USA

    William A. Stein

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  1. William A. Stein
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Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    John E. Cremona

  2. Facultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Pau Gargallo, 5, 08028, Barcelona, Spain

    Joan-Carles Lario  & Jordi Quer  & 

  3. Department of Mathematics, University of California, M/C 3840, Berkeley, CA, 94720-3840, USA

    Kenneth A. Ribet

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Stein, W.A. (2004). Shafarevich-Tate Groups of Nonsquare Order. In: Cremona, J.E., Lario, JC., Quer, J., Ribet, K.A. (eds) Modular Curves and Abelian Varieties. Progress in Mathematics, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7919-4_17

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  • DOI: https://doi.org/10.1007/978-3-0348-7919-4_17

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9621-4

  • Online ISBN: 978-3-0348-7919-4

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