By Ellen E. Eischen, Ling Long, Rachel Pries, Katherine E. Stange
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Extra info for Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop
Then is the unique (up to scaling) non-trivial anticyclotomic p-adic idele class character. S. Balakrishnan et al. 6. Note that W A =K ! 1. K/ and compute the kernel of this pairing. 7. 5. 5. For any finite prime w of K, the natural inclusion Kw ,! A induces a map Ãw W Kw ! K/, and we write w D ı Ãw . P/ as a local denominator of P at w. P/ follows from the Weierstrass equation for E and the fact that Ow is a principal ideal domain. Finally, we let denote the -adic -function of E. P/ D . P// . P/.
K/ ! Qp . The kernel of such a (non-zero) Qp -linear map corresponds to a Zp -extension of K. In particular, the anticyclotomic Zp -extension of K corresponds to a Qp -linear , where c denotes complex conjugation. K/ ! Qp such that ı c D The resulting anticyclotomic p-adic height pairing is denoted by . ; / . One key step of our work is an explicit description of the map , see Sect. 2. P; P/ . Mazur et al. P/ D . P// . P//; w−p1 where is one of the prime divisors of p in K and the remaining notation is defined in Sect.
7 G14 F14 ! 7 G13 F11 7! G11 F5 7! G6 C5 7! C6 F6 7! G6 C6 7! e D6 F7 7! G7 F9 7! G9 F14 7! G14 F5 7! G5 C5 7! e D5 C7 ! 7 D7 C9 ! 7 D9 F2 $ F3 C2 $ C3 F10 7! G10 F1 $ G1 Action on Pic Y a C1 7! D1 D1 $ e C1 C2 7! e C2 C3 7! e C3 Action on Pic X a P3 $ P4 Q1 $ Q3 Q2 $ Q4 Action on Weierstrass points i c q p 10 ab c p 10 ab p a Á0 Â0 7! Â0 ÂiC 7! Âi 7! q p a 7! Á0 7! p 2C2 2 q p 7! 2C2 2 p p c 7! c q i 7! Action on splitting field C8 $ C9 C6 7! D6 C5 7! D5 C6 7! D6 C5 7! D5 C4 ! 7 D4 C6 ! 7 D6 C7 7!
Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop by Ellen E. Eischen, Ling Long, Rachel Pries, Katherine E. Stange