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The Manin constant in the semistable case
For an optimal modular parametrization $J_{0}(n){\twoheadrightarrow}E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$ , Manin conjectured the agreement of two natural $\mathbb{Z}$Expand
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Poitou-Tate without restrictions on the order
The Poitou-Tate sequence relates Galois cohomology with restricted ramification of a finite Galois module $M$ over a global field to that of the dual module under the assumption that $\#M$ is a unitExpand
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Selmer groups as flat cohomology groups
Given a prime number $p$, Bloch and Kato showed how the $p^\infty$-Selmer group of an abelian variety $A$ over a number field $K$ is determined by the $p$-adic Tate module. In general, theExpand
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The $A_{\text{inf}}$ -cohomology in the semistable case
For a proper, smooth scheme $X$ over a $p$ -adic field $K$ , we show that any proper, flat, semistable ${\mathcal{O}}_{K}$ -model ${\mathcal{X}}$ of $X$ whose logarithmic de Rham cohomology isExpand
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The $p$-parity conjecture for elliptic curves with a $p$-isogeny
For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-WeilExpand
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TOPOLOGY ON COHOMOLOGY OF LOCAL FIELDS
Arithmetic duality theorems over a local field $k$ are delicate to prove if $\text{char}\,k>0$. In this case, the proofs often exploit topologies carried by the cohomology groups $H^{n}(k,G)$ forExpand
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The ℓ-parity conjecture over the constant quadratic extension
Abstract For a prime ℓ and an abelian variety A over a global field K, the ℓ-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton–Dyer, the ℤℓ-corank of the ℓ∞-SelmerExpand
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Selmer groups and class groups
Abstract Let $A$ be an abelian variety over a global field $K$ of characteristic $p\geqslant 0$. If $A$ has nontrivial (respectively full) $K$-rational $l$-torsion for a prime $l\neq p$, we exploitExpand
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Local factors valued in normal domains
We give an exposition of Deligne’s theory of local ϵ0-factors over fields and discrete valuation rings under the assumption that the theory over the complex numbers is known. We then employ standardExpand
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