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MODULAR FORM CONGRUENCES AND SELMER GROUPS

Published online by Cambridge University Press:  25 March 2003

WILLIAM J. McGRAW  and
KEN ONO
WILLIAM J. McGRAW
Affiliation:
Department of Mathematics, University of Wisconsin, Madison WI 53706, USAmcgraw@math.wisc.edu
KEN ONO
Affiliation:
Department of Mathematics, University of Wisconsin, Madison WI 53706, USAono@math.wisc.edu
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Abstract

Motivated by Cremona and Mazur's notion of visibility of elements in Shafarevich–Tate groups [6, 27], there have been a number of recent works which test its compatibility with the Birch and Swinnerton–Dyer conjecture and the Bloch–Kato conjecture. These conjectures provide formulas for the orders of Shafarevich–Tate groups in terms of values of $L$-functions. For example, one may see recent work of Agashe, Dummigan, Stein and Watkins [1, 2, 10, 11]. In their examples, they find that the presence of visible elements agrees with the expected divisibility properties of the relevant $L$-values.

Keywords

11F3311F3711F6711G40
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 67 , Issue 2 , April 2003 , pp. 302 - 318
Copyright
The London Mathematical Society 2003

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