The <i>p</i>-parity conjecture for elliptic curves with a <i>p</i>-isogeny

Česnavičius, Kęstutis

doi:10.1515/crelle-2014-0040

Cited by 12 publications

(20 citation statements)

References 27 publications

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“…It remains to prove the equivalence of (1), (2) and (3). Now Ω = Ω if and only if ord p ( Ω Ω ) is even, and we can relate this to the parity of the analytic rank and of the p ∞ -Selmer rank of E/Q: 3) or nonsplit multiplicative reduction (as ker φ ∼ = Z/pZ or μ p in the split multiplicative case, and it has no points after an unramified quadratic twist).…”

Section: Periods Of Elliptic Curves Over Qmentioning

confidence: 99%

“…1.4]) and Cassels' formula for the parity of the p ∞ -Selmer rank for an elliptic curve with a p-isogeny([7, Rmk. is odd if and only if E has split multiplicative reduction at l. So the left-hand side in(2) is the number of primes of split multiplicative reduction. is odd if and only if E has split multiplicative reduction at l. So the left-hand side in(2) is the number of primes of split multiplicative reduction.…”

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confidence: 99%

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Local invariants of isogenous elliptic curves

Dokchitser

2014

Trans. Amer. Math. Soc.

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Section: Periods Of Elliptic Curves Over Qmentioning

confidence: 99%

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confidence: 99%

Local invariants of isogenous elliptic curves

Dokchitser

2014

Trans. Amer. Math. Soc.

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“…If

\bar{T}

is reducible, one applies a different argument based on a combination of an Euler characteristic formula for Breuil–Kisin modules (equipped with tame descent data) with a generalisation of a formula of Cassels [, Theorem 1.1]. This yields the following abstract result (see [, Theorem 2; , Theorem 5.7; , Theorem 1.4; , Theorem 2.1] for analogous results for Selmer groups of elliptic curves and abelian varieties, respectively). Theorem If, in the global situation discussed before Theorem (with

p \neq 2

), the following conditions are satisfied: (i)

\bar{T}

contains a

Γ_{L}

‐stable Lagrangian subspace; (ii)for each finite prime

v ∤ p

L

there exist a finite abelian extension

L false(v false) / L_{v}

and a finite Galois extension

K (v) / L (v)

such that

p ∤ [K (v) : L (v)]

and

{T false|}_{{normalΓ}_{K (v)}}

is unramified; (iii)for each finite prime

v ∣ p

L

there exists a finite abelian extension

K false(v false) / L_{v}

such that

{T |}_{Γ_{K false(v false)}} = T_{π} G

for some

p

‐divisible group

G

(equipped with an

scriptO

‐action) over

O_{K (v)<...>}

…”

Section: Introductionmentioning

confidence: 99%

Compatibility of arithmetic and algebraic local constants, II: the tame abelian potentially Barsotti-Tate case

Nekovář

2017

Proc. London Math. Soc.

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We prove the compatibility of arithmetic local constants of Mazur and Rubin with the usual local constants for pairs of congruent self-dual Galois representations that become Barsotti-Tate over a tamely ramified abelian extension. This allows us to complete the proof of the p-parity conjecture (p > 2) for Selmer groups of Hilbert modular forms of parallel weight 2 and abelian varieties with real multiplication (in particular, elliptic curves) over totally real number fields. IntroductionLet L be a finite extension of Q , let K be a finite extension of Q p (where p > 2) with ring of integers O, and let V be a finite-dimensional vector space over K equipped with a continuous K-linear action of Γ L = Gal(L/L). Assume that V is self-dual in the sense that there exists a Γ L -equivariant nondegenerate skew-symmetric pairing ,is a uniformiser) has coefficients in the finite field F := O/πO and the skew-symmetric pairing , :representation of the Weil-Deligne group of L attached to V (respectively to D pst (V ), [26; 27, I.1.3.2]) if = p (respectively if = p). Let ψ be a nontrivial additive character of L and dx ψ the Haar measure on L that is self-dual with respect to ψ. The local constant ε(V ) := ε(W D(V ), ψ, dx ψ ) does not depend on ψ and is equal to ±1 [37, Proposition 2.2.1]. Mazur and Rubin expect their arithmetic local constant δ L (T, T ) to be related to ε(V )/ε(V ). This was checked by explicit calculations in certain cases arising from abelian varieties [17]. It was subsequently proved [41, Theorem 2.17] thatholds in full generality if = p. One expects this relation to be true even if = p, provided the Hodge-Tate weights of V and V are the same (including the multiplicities). This was verified in two very special cases in [41, § 3].The main local result of the present article is the following.Theorem A (= Theorem 5.5). If = p = 2 and if there exist abelian tamely ramified finite extensions K/L and K /L and p-divisible groups (equipped with an O-action JAN NEKOVÁŘtogether with the existence of a cohomological version of the Cassels-Tate pairing constructed by Flach [24, Theorem 1] imply that Theorem 2.17] and for v | p in Theorem A imply the following result (the assumption of tame ramification is no longer necessary, since the statement is invariant under base change to an arbitrary abelian extension of odd degree).Theorem B (= Theorem 6.7). Assume that, in the global situation considered above, for every prime v | p of L there exist finite abelian extensionswhere ε(V ) := v ε v (V ) and ε(V ) := v ε v (V ) (the product is taken over all primes of L and ε v (V ) = ε v (V ) := (−1) dimK(V )/2 for all v | ∞).

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“…In the number field case, this conjecture is now known in several cases, in particular when A is an elliptic curve and the ground field is K = Q by the work of the Dokchitser brothers [DD08, DD09, DD10, DD11], Kim [Kim07,Kim09], Nekovář [Nek13,Nek01,Nek09], [Nek06,Ch. 12],Česnavičius [Ces12], Coates et al [CFKS10] and others.…”

Section: Introductionmentioning

confidence: 99%

The -parity conjecture for abelian varieties over function fields of characteristic

Trihan

Yasuda

2014

Compositio Math.

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Let A/K be an abelian variety over a function field of characteristic p > 0 and let be a prime number ( = p allowed). We prove the following: the parity of the corank r of the -discrete Selmer group of A/K coincides with the parity of the order at s = 1 of the Hasse-Weil L-function of A/K. We also prove the analogous parity result for pure -adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin-Tate conjecture.

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The p-parity conjecture for elliptic curves with a p-isogeny

Abstract: For an elliptic curve

Cited by 12 publications

References 27 publications

Local invariants of isogenous elliptic curves

Local invariants of isogenous elliptic curves

Compatibility of arithmetic and algebraic local constants, II: the tame abelian potentially Barsotti-Tate case

The -parity conjecture for abelian varieties over function fields of characteristic

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