dc.contributor.advisor Kisin, Mark en_US dc.contributor.author Tang, Yunqing en_US dc.date.accessioned 2017-07-25T14:42:17Z dc.date.created 2016-05 en_US dc.date.issued 2016-05-04 en_US dc.date.submitted 2016 en_US dc.identifier.citation Tang, Yunqing. 2016. Algebraicity Criteria and Their Applications. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. en_US dc.identifier.uri http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493480 dc.description.abstract We use generalizations of the Borel–Dwork criterion to prove variants of the Grothedieck–Katz p-curvature conjecture and the conjecture of Ogus for some classes of abelian varieties over number fields. The Grothendieck–Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for all but finitely many primes p, has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on P^1 − {0, 1, ∞}. We prove a variant of this conjecture for P^1 − {0, 1, ∞}, which asserts that if the equation satisfies a certain convergence condition for all p, then its monodromy is trivial. For those p for which the p-curvature makes sense, its vanishing implies our condition. We deduce from this a description of the differential Galois group of the equation in terms of p-curvatures and certain local monodromy groups. We also prove similar variants of the p-curvature conjecture for a certain elliptic curve with j-invariant 1728 minus its identity and for P^1 − {±1, ±i, ∞}. Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of l-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. We confirm Ogus’ conjecture for some classes of abelian varieties, under the assumption that these cycles lie in the Betti cohomology with real coefficients. These classes include abelian varieties of prime dimension that have nontrivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture. We also discuss some strengthenings of the theorem of Bost. en_US dc.description.sponsorship Mathematics en_US dc.format.mimetype application/pdf en_US dc.language.iso en en_US dash.license LAA en_US dc.subject Mathematics en_US dc.title Algebraicity Criteria and Their Applications en_US dc.type Thesis or Dissertation en_US dash.depositing.author Tang, Yunqing en_US dc.date.available 2017-07-25T14:42:17Z thesis.degree.date 2016 en_US thesis.degree.grantor Graduate School of Arts & Sciences en_US thesis.degree.level Doctoral en_US thesis.degree.name Doctor of Philosophy en_US dc.contributor.committeeMember Gross, Benedict en_US dc.contributor.committeeMember Mazur, Barry en_US dc.type.material text en_US thesis.degree.department Mathematics en_US dash.identifier.vireo http://etds.lib.harvard.edu/gsas/admin/view/864 en_US dc.description.keywords algebraicity criteria; Grothendieck--Katz conjecture; abelian varieties en_US dash.author.email yqtang1989@gmail.com en_US dash.contributor.affiliated Tang, Yunqing
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