Problem 8.8.2
Cristian Wuthrich and Stein (mostly Wuthrich) have written a bunch
of code related to using Peter Schneider's work on
![$ p$](img3.png)
-adic
analogues of the BSD conjecture to compute
![$ \char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$](img401.png)
at certain
primes where the methods of Kolyvagin and Kato fail.
Remark 8.8.3 (From Christian Wuthrich.)
Note that the paper mentioned above, as
far as I have written it is, to my taste, more or less done. I
should add some data of numerical results which you can of course
ask the students to produce. But there is no need or interest for a
long list. I have not written yet the introduction nor the part I
named technical details (but I am not sure if I actually want to do
that).
Of course, I am very happy that part (or the whole of) shark will be
included in SAGE.
Remark 8.8.4 (From Christian Wuthrich.)
Schneider's (and simultanoeously
Perrin-Riou's work) is strictly speaking not on the p-adic BSD. The
most important result to use is Kato's which links the algebraic to
the analytic side. Look in the article we write together for a
tigher bound in the case
![$ L(E,1)$](img402.png)
is not zero. Your katobound in
sage is not sharp.