𝑝-primary parts Sample Clauses

𝑝-primary parts. In this subsection we briefly outline a useful decomposition of the conjectures eTNC𝐸/𝐾 and RS𝑉 𝐸/𝐾 into 𝑝-primary parts. To do this we fix a prime number 𝑝 and an isomorphism 𝑝 . Using × Stickelberger element 𝜃 (𝑟) as an element of 𝑝 [𝐸 ]. this isomorphism we may identify ˆ𝐸 with Hom(𝐸, Q𝑝 ) and can therefore view the 𝑟𝑡ℎ-order M : d We now consider the𝐸r/e𝐾g,𝑈u,lΣator isomorphism P𝐸/𝐾,𝑈 ,𝑝 𝑝 [𝐸 ] (𝐶𝐸,𝑈 ,Σ ⊗Z 𝑝 ) → 𝑝 [𝐸 ] M which is defined in the same fashion as P𝐸/𝐾,𝑈 . Using this map we can then define the 𝑝-primary part of the zeta element 𝔷𝐸/𝐾,𝑈 ,Σ: Definition 2.8.N. We define 𝔷𝐸/𝐾,𝑈 ,Σ,𝑝 to be the unique element of 𝑝 ⊗Z𝑝 dZ𝑝 [𝐸 ] (𝐶𝐸,𝑈 ,Σ ⊗Z Z𝑝 ) that maps under P𝐸,𝑈 ,Σ,𝑝 to give 𝜃 (0) . 𝐸/𝐾,𝑈 ,Σ Given these constructions we then have the 𝑝-part of the conjecture eTNC𝐸/𝐾 : M Conjecture 2.8.8 (eTNC𝐸/𝐾,𝑈 ,Σ,𝑝 ). There is an equality M dZ𝑝 [𝐸 ] (𝐶𝐸,𝑈 ,Σ ⊗Z Z𝑝 ) = (𝔷𝐸/𝐾,𝑈 ,Σ,𝑝 )Z𝑝 [𝐸 ] inside 𝑝 ⊗Z𝑝 dZ𝑝 [𝐸 ] (𝐶𝐸,𝑈 ,Σ ⊗Z Z𝑝 ). eTNC𝐸/𝐾,𝑈 ,Σ is equivalent to the conjunction 𝑝 prime eTNC𝐸/𝐾,𝑈,Σ,𝑝 . It is a straightforward exercise comparing the maps P𝐸,𝑈 ,Σ,𝑝 and P𝐸,𝑈 ,Σ ⊗Z 𝑝 to verify that