The equivariant Tamagawa number conjecture and modular symbols

Bley, Werner

doi:10.1007/s00208-012-0837-6

Cited by 8 publications

(8 citation statements)

References 10 publications

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“…• In [4], it is shown that for each elliptic curve A/Q with L(A/Q, 1) = 0 there are infinitely many primes p and for each such prime p infinitely many (cyclic) p-extensions F/Q such that C p (A, Z[Gal(F/Q)]) holds. All of these examples satisfy our hypotheses and are such that A(F ) p vanishes.…”

Section: Verifications Of Conjecture C P (A Z[g])mentioning

confidence: 99%

“…This limitation of our previous methods is consistent with those occurring in all existing verifications of p-components of the eTNC for any elliptic curves. Indeed, in the settings of the theoretical verifications obtained by the first author in [3], by Burns, Wuthrich and the second author in [9], or of the recent extensions of these results by Burns and the second author in [8]; as well as of the numerical verifications carried out both in [9] and in our previous article [4]; a full verification of this conjecture was only ever achieved in situations which forced the Z p [G]-module A(F ) p to be projective. On the other hand, even for n = 1 (meaning that the extension F/k has degree p), the result [8,Thm.…”

Section: Introductionmentioning

confidence: 98%

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Congruences for critical values of higher derivatives of twisted Hasse–WeilL-functions

Bley

Castillo

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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The main aim of the work is to assess physical parameters of forest woodchips and their impact on the prices achieved by the supplier in transactions with a power plant. During fragmentation of logging residue, high content of green matter and contaminants negatively impacts the quality parameters that serve as basis for settlements. The analysis concerns data on the main parameters -water content, fuel value, sulphur and ash content -from 252 days of deliveries of forest chips to a power plant. The deliveries were realised from forested areas on an average about 340 km from the plant. Average water content and the resultant fuel value of forest chips was within 27-47% and 8.7-12.9 GJ×Mg −1 (appropriately), respectively. They depend on the month in which they are delivered to the power plant. The threshold values for the above-mentioned parameters are set by the plant at a real level and the suppliers have no problems with meeting them. The parameter that is most frequently exceeded is ash content (11.5% of cases). The settlement system does not differentiate on the basis of the transport distance but gives possibility to lower the settlement price when the quality parameters are not met but provides no reward for deliveries with parameters better than the average ones. On the basis of results obtained, it was calculated that average annual settlement price is lower than the contract price by about 0.20 PLN×GJ −1 , which in case of the analysed company may translate into an average daily loss of about 700 PLN.

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Section: Verifications Of Conjecture C P (A Z[g])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Congruences for critical values of higher derivatives of twisted Hasse–WeilL-functions

Bley

Castillo

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…In this section we explain how one may numerically compute the Mazur-Tate pairing (5). The computation can be reduced to the computation of local Tate duality pairings which, in turn, may in simple situations be computed by the evaluation of Hilbert symbols thanks to recent results of Fisher and Newton [18] or of Visse [30].…”

Section: Computation Of the Mazur-tate Pairingmentioning

confidence: 99%

“…This limitation of our previous methods is consistent with those occurring in all existing verifications of p-components of the eTNC for any elliptic curves. Indeed, in the settings of the theoretical verifications obtained by the first author in [5], by Burns, Wuthrich and the second author in [11], or of the recent extensions of these results by Burns and the second author in [10], as well as of the numerical verifications carried out in [3,4,11] and in our previous paper [6], a full verification of this conjecture was only ever achieved in situations which forced the Z p [G]-module A(F) p to be projective.…”

Section: Introductionmentioning

confidence: 98%

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

Bley¹,

Castillo²

2021

Math. Proc. Camb. Phil. Soc.

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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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“…In recent years, there has been much interest in the formulation and study of integral refinements of Deligne's Period Conjecture for the equivariant L-values that are associated to the base change of an abelian variety through a Galois extension of number fields. We refer the reader to [1,2,3,4,5,11,12,15,20]. However, as far as we are aware, any theoretical or numerical evidence obtained for such refinements has been restricted to the case of elliptic curves.…”

Section: Introductionmentioning

confidence: 99%

Numerical Evidence for a refinement of Deligne's Period Conjecture for Jacobians of Curves

Evans¹,

Castillo²,

Wiersema³

2023

Preprint

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Let A/Q be a Jacobian variety and let F be a totally real, tamely ramified, abelian number field. Given a character ψ of F/Q, Deligne's Period Conjecture asserts the algebraicity of the suitably normalised value L(A, ψ, 1) at z = 1 of the Hasse-Weil-Artin L-function of the ψtwist of A. We formulate a conjecture regarding the integrality properties of the family of normalised L-values (L(A, ψ, 1)) ψ , and its relation to the Tate-Shafarevich group of A over F . We numerically investigate our conjecture through p-adic congruence relations between these values.

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The equivariant Tamagawa number conjecture and modular symbols

Cited by 8 publications

References 10 publications

Congruences for critical values of higher derivatives of twisted Hasse–WeilL-functions

Congruences for critical values of higher derivatives of twisted Hasse–WeilL-functions

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

Numerical Evidence for a refinement of Deligne's Period Conjecture for Jacobians of Curves

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