Selmer群的重数与Artin扭的根数
Multiplicities in Selmer groups and root numbers for Artin twists
摘要：
令$K/F$是数域的有限伽罗瓦扩张，$σ$是$\mathrm{Gal}(K/F)$的绝对不可约、自对偶表示。设$p$是奇素数，考虑两条椭圆曲线$E_1, E_2$，它们在$p$以上的素数处具有良好的普通约化，并具有等价的mod- $p$ Galois表示.本文研究了在$K$上的$E_i$的$p^\infty$ -Selmer群的表示空间中$σ$的重数的奇偶性的变化.我们还比较了扭转的$E_i/F$由$σ$的根数，并表明$p$ -奇偶猜想成立的扭转的$E_1/F$由$σ$的当且仅当它成立的扭转的$E_2/F$由$σ$。我们还表示Mazur-Rubin-Nekovásness的算术局部常数在某些地方岩泽不变量。
英文摘要：
Let $K/F$ be a finite Galois extension of number fields and $σ$ be an absolutely irreducible, self-dual representation of $\mathrm{Gal}(K/F)$. Let $p$ be an odd prime and consider two elliptic curves $E_1, E_2$ with good, ordinary reduction at primes above $p$ and equivalent mod-$p$ Galois representations. In this article, we study the variation of the parity of the multiplicities of $σ$ in the representation space associated to the $p^\infty$-Selmer group of $E_i$ over $K$. We also compare the root numbers for the twist of $E_i/F$ by $σ$ and show that the $p$-parity conjecture holds for the twist of $E_1/F$ by $σ$ if and only if it holds for the twist of $E_2/F$ by $σ$. We also express Mazur-Rubin-Nekovář's arithmetic local constants in terms of certain local Iwasawa invariants.
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