The -parity conjecture for abelian varieties over function fields of characteristic
2014 ◽
Vol 150
(4)
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pp. 507-522
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AbstractLet $A/K$ be an abelian variety over a function field of characteristic $p>0$ and let $\ell $ be a prime number ($\ell =p$ allowed). We prove the following: the parity of the corank $r_\ell $ of the $\ell $-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 $\ell $-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.
1995 ◽
Vol 38
(2)
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pp. 167-173
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2017 ◽
Vol 2019
(14)
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pp. 4469-4515
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2014 ◽
Vol 10
(03)
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pp. 705-735
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2018 ◽
Vol 2018
(741)
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pp. 133-159
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2020 ◽
Vol 16
(05)
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pp. 1081-1109
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2013 ◽
Vol 09
(05)
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pp. 1249-1262
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1998 ◽
Vol 09
(08)
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pp. 1041-1066
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