We then start out on the proof by using Coleman's amazing idea of representing norm-compatible systems of units by power series. These allow us to do weird things like take 'logarithmic derivatives' of a unit, which miraculously when applied to a tower of cyclotomic units ends up constructing the p-adic zeta function. We end by proving Iwasawa's relatively classical theorem that the Iwasawa module of local units modulo the closure of cyclotomic units "satisfies the main conjecture.
Seminar on the Gross-Zagier formula and Kolyvagin's work
We remark that this procedure should work in a similar manner for other Euler systems one should get a "Coleman map" associating to an Euler system a p-adic L function. We will also explain how via class field theory one would then be able to deduce the actual main conjecture from a statement similar to that which Rong proved last time, setting the stage for Yihang to present Rubin's slick Euler system argument next time to once again prove that two things that are conjecturally zero are at least suitably equal.
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Iwasawa's Main Conjecture reveals a deep relation between the p-adic zeta function and the class groups of cyclotomic fields as Galois modules. This conjecture is under the general philosophy that there should be a correspondence between p-adic analytic invariants and arithmetic invariants. To go from the zeta side to the arithmetic side, we saw in Tom's talk last week the improtant Theorem of Iwasawa that relates the p-adic zeta function to the cyclotomic units.
Thus in order to prove the Main Conjecture, we need to compare the cyclotomic units and the class groups of cyclotomic fields, which are now in the same arithmetic world.
The cyclotomic units can be regarded as arithmetic incarnations of zeta, which is reflected in the fact that they form a very special family, namely an Euler system. In this talk we will study the cyclotomic units from this point of view.
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Heegner points and rankin l series pdf
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Heegner Points and Rankin L-Series
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Read more about accessing full-text. This article studies a distinguished collection of so-called generalized Heegner cycles in the product of a Kuga—Sato variety with a power of a CM elliptic curve.
Duke Mathematical Journal
Its main result is a p -adic analogue of the Gross—Zagier formula which relates the images of generalized Heegner cycles under the p -adic Abel—Jacobi map to the special values of certain p -adic Rankin L -series at critical points that lie outside their range of classical interpolation.
Source Duke Math. Zentralblatt MATH identifier
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