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Concept# P-adic number

Summary

In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number p rather than ten, and extending (possibly infinitely) to the left rather than to the right. Formally, given a prime number p, a p-adic number can be defined as a series
:s=\sum_{i=k}^\infty a_i p^i = a_k p^k + a_{k+1} p^{k+1} + a_{k+2} p^{k+2} + \cdots
where k is an integer (possibly negative), and each a_i is a integer such that 0\le a_i < p. A p-adic integer is a p-adic number such that k\ge 0.
In general the series that represents a p-adic number is not convergent in the usual sense, but it is convergent

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We establish p-adic versions of the Manin-Mumford conjecture, which states that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a p-adic field or its ring of integers, respectively. In particular, we show that the underlying rigidity results for algebraic functions generalize to suitable p-adic analytic functions. This leads us to uncover purely p-adic Manin-Mumford-type results for formal groups not coming from abelian schemes. Moreover, we observe that a version of the Tate-Voloch conjecture holds: torsion points either lie squarely on a subscheme or are uniformly bounded away from it in the p-adic distance.

We define and study in terms of integral IwahoriâHecke algebras a new class of geometric operators acting on the Bruhat-Tits building of connected reductive groups over p-adic fields. These operators, which we call U-operators, generalize the geometric notion of "successors" for trees with a marked end. The first main contributions of the thesis are:
(i) the integrality of the U-operator over the spherical Hecke algebra using the compatibility between Bernstein and Satake homomorphisms,
(ii) in the unramified case, the U-operator attached to a cocharacter is a right root of the corresponding Hecke polynomial.
In the second part of the thesis, we study some arithmetic aspects of special cycles on (products of) unitary Shimura varieties, these cycles are expected to yield new results towards the BlochâBeilinson conjectures. As a global application of (ii), we obtain:
(iii) the horizontal norm relations for these GGP cycles for arbitrary n, at primes where the unitary group splits.
The general local theory developed in the first part of the thesis, has the potential to result in a number of global applications along the lines of (iii) (involving other Shimura varieties and also vertical norm relations) and offers
new insights into topics such as the BlasiusâRogawski conjecture as well.

We construct "generalized Heegner cycles" on a variety fibered over a Shimura curve, defined over a number field. We show that their images under the p-adic Abel-Jacobi map coincide with the values (outside the range of interpolation) of a p-adic L-function L-p(f, chi) which interpolates special values of the Rankin-Selberg convolution of a fixed newform f and a theta-series theta(chi) attached to an unramified Hecke character of an imaginary quadratic field K. This generalizes previous work of Bertolini, Darmon, and Prasanna, which demonstrated a similar result in the case of modular curves. Our main tool is the theory of Serre-Tate coordinates, which yields p-adic expansions of modular forms at CM points, replacing the role of q-expansions in computations on modular curves.