Congruences for critical values of higher derivatives of twisted Hasse–Weil L-functions, III

Let A be an abelian variety defined over a number field k, let p be an odd prime number and let $F/k$ be a cyclic extension of p-power degree. Under not-too-stringent hypotheses we give an interpretation of the p-component of the relevant case of the equivariant Tamagawa number conjecture in terms o...

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Published inMathematical proceedings of the Cambridge Philosophical Society Vol. 173; no. 2; pp. 431 - 456
Main Authors BLEY, WERNER, MACIAS CASTILLO, DANIEL
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.09.2022
Subjects
Algebra
Congruences
Equality
Hypotheses
Mathematical analysis
Number theory
Numerical analysis
Prime numbers
11G35
11G05
11G40
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Summary:Let A be an abelian variety defined over a number field k, let p be an odd prime number and let $F/k$ be a cyclic extension of p-power degree. Under not-too-stringent hypotheses we give an interpretation of the p-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of A of the ${\rm Gal}(F/k)$ -valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in situations in which the p-completion of the Mordell–Weil group of A over F is not a projective Galois module.
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ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004121000657