Zero cycles on abelian varieties, Somekawa K-groups and local symbols /
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| Author / Creator: | Gazaki, Evangelia, author. |
|---|---|
| Imprint: | 2016. Ann Arbor : ProQuest Dissertations & Theses, 2016 |
| Description: | 1 electronic resource (110 pages) |
| Language: | English |
| Format: | E-Resource Dissertations |
| Local Note: | School code: 0330 |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/11674564 |
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| 100 | 1 | |a Gazaki, Evangelia, |e author. | |
| 245 | 1 | 0 | |a Zero cycles on abelian varieties, Somekawa K-groups and local symbols / |c Gazaki, Evangelia. |
| 260 | |c 2016. | ||
| 264 | 1 | |a Ann Arbor : |b ProQuest Dissertations & Theses, |c 2016 | |
| 300 | |a 1 electronic resource (110 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 500 | |a Advisors: Kazuya Kato Committee members: Madhav Nori. | ||
| 502 | |b Ph.D. |c University of Chicago, Division of the Physical Sciences, Department of Mathematics |d 2016. | ||
| 510 | 4 | |a Dissertation Abstracts International, |c Volume: 77-12(E), Section: B. | |
| 520 | |a This thesis consists of two main parts. The first part concerns zero cycles on abelian varieties and their relation to some Milnor type K-groups. In chapter 1 we recall some basic properties of Milnor K-groups and their generalizations, the Somekawa K-groups. The main result of the first part is presented in chapter 2, where we construct, for an abelian variety A over a field k, a decreasing filtration {Fr}r≥0 of the group CH0(A) having the property that the successive quotients Fr/F r+1 are isomorphic after [special characters omitted] to a Somekawa type K-group. We then focus on the case when the base field is a finite extension of Q p. Using the above filtration, we prove some results of arithmetic interest about the structure of the albanese kernel, the kernel of the cycle map to etale cohomology and the Brauer-Manin pairing. The results of this chapter are gathered in one paper, [14]. | ||
| 520 | |a Chapter 3 serves as a bridge between the first and the second part of this thesis. In this chapter we work with smooth quasi-projective varieties, introducing Suslin's singular homology group and Wiesend's tame class group. The latter group is a first generalization in higher dimensions of the generalized Jacobian varieties of a smooth projective curve. Using these two geometric invariants, we generalize the main theorem of chapter 2 for semiabelian varieties. We close the chapter by providing some motivation towards a more general reciprocity theory. | ||
| 520 | |a The second part concerns a newly developed theory about reciprocity functors introduced by Ivorra and Rulling in [20]. This theory generalizes the theory of Rosenlicht-Serre about local symbols on commutative algebraic groups. In particular, we will see that every reciprocity functor M has local symbols corresponding to any smooth complete curve C over a field k. These local symbols induce a complex (C). In chapter 4 we focus on the case of a smooth complete curve C over an algebraically closed field k and we compute under two assumptions the homology of the local symbol complex in terms of K-groups of reciprocity functors. We then close the thesis by providing important examples where the assumptions are satisfied. The results of this chapter are gathered in one paper, [15]. | ||
| 546 | |a English | ||
| 590 | |a School code: 0330 | ||
| 690 | |a Mathematics. | ||
| 710 | 2 | |a University of Chicago. |e degree granting institution. | |
| 720 | 1 | |a Kazuya Kato |e degree supervisor. | |
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| 856 | 4 | 0 | |u https://dx.doi.org/10.6082/M14M92GS |y Knowledge@UChicago |
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