We study two conjectures introduced by Flach and Morin in [FM18] for schemes over a perfect field of characteristic p > 0. The first conjecture relates a p-adic extension of the \u00e9tale motivic cohomology with compact support on general schemes introduced by Geisser in [Gei06] to rigid cohomology with compact support, and is proved here. The second, relates a p-adic Borel-Moore motivic homology with the dual of rigid cohomology with compact support, and is proved in the smooth case. For this, we also prove a generalization of the comparison theorem from rigid cohomology to overconvergent de Rham-Witt cohomology in [DLZ11].
", "doi": "10.7907/DCJJ-E164", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:8863", "collection": "thesis", "collection_id": "8863", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05182015-134458833", "type": "thesis", "title": "On the Construction of Higher \u00e9tale Regulators", "author": [ { "family_name": "Fan", "given_name": "Sin Tsun Edward", "clpid": "Fan-Sin-Tsun-Edward" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We present three approaches to define the higher \u00e9tale regulator maps \u03a6r,net : Hret(X,Z(n)) → HrD(X,Z(n)) for regular arithmetic schemes. The first two approaches construct the maps on the cohomology level, while the third construction provides a morphism of complexes of sheaves on the \u00e9tale site, along with a technical twist that one needs to replace the Deligne-Beilinson cohomology by the analytic Deligne cohomology inspired by the work of Kerr, Lewis, and M\u00fcller-Stach. A vanishing statement of infinite divisible torsions under \u03a6r,net is established for r > 2n + 1.", "doi": "10.7907/Z9BZ63Z1", "publication_date": "2015", "thesis_type": "phd", "thesis_year": "2015" }, { "id": "thesis:8427", "collection": "thesis", "collection_id": "8427", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05292014-153502602", "primary_object_url": { "basename": "thesis_final.pdf", "content": "final", "filesize": 506804, "license": "other", "mime_type": "application/pdf", "url": "/8427/1/thesis_final.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "On the Local Tamagawa Number Conjecture for Tate Motives", "author": [ { "family_name": "Daigle", "given_name": "Gerald Joseph III (Jay)", "clpid": "Daigle-Gerald-Joseph-III" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Hedayat Zadeh Razavi", "given_name": "S. Mohammad Hadi", "clpid": "Hedayat-Zadeh-Razavi-S-M-H" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "There is a wonderful conjecture of Bloch and Kato that generalizes both the analytic Class Number Formula and the Birch and Swinnerton-Dyer conjecture. The conjecture itself was generalized by Fukaya and Kato to an equivariant formulation. In this thesis, I provide a new proof for the equivariant local Tamagawa number conjecture in the case of Tate motives for unramified fields, using Iwasawa theory and (\u03c6,\u0393)-modules, and provide some work towards extending the proof to tamely ramified fields.", "doi": "10.7907/RFXG-4E72", "publication_date": "2014", "thesis_type": "phd", "thesis_year": "2014" }, { "id": "thesis:6756", "collection": "thesis", "collection_id": "6756", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:12212011-095330433", "type": "thesis", "title": "On the Weil-\u00e9tale Cohomology of S-Integers", "author": [ { "family_name": "Chiu", "given_name": "Yi-Chih", "clpid": "Chiu-Yi-Chih" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Jorza", "given_name": "Andrei", "clpid": "Jorza-A" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We generalize the Lichtenbaum's prototype of Weil-\u00e9tale cohomology to S-integers and study its relation to the Tate sequences. In the final part, we present a more natural way to define Weil-\u00e9tale cohomology for one-dimensional arithmetic schemes motivated by a dual quasi-isomorphism between Weil-\u00e9tale cohomology and \u00e9tale cohomology. ", "doi": "10.7907/W4CA-6489", "publication_date": "2012", "thesis_type": "phd", "thesis_year": "2012" }, { "id": "thesis:7090", "collection": "thesis", "collection_id": "7090", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05292012-142939643", "primary_object_url": { "basename": "Thesis.pdf", "content": "final", "filesize": 361309, "license": "other", "mime_type": "application/pdf", "url": "/7090/1/Thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "One the P-Adic Local Invariant Cycle Theorem", "author": [ { "family_name": "Wu", "given_name": "Yi-Tao", "clpid": "Wu-Yi-Tao" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Jorza", "given_name": "Andrei", "clpid": "Jorza-A" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The aim of this paper is to consider the $p$-adic local invariant cycle theorem in the mixed characteristic case.
\r\n\r\nIn the first part of the paper, via case-by-case discussion, we construct the $p$-adic specialization map, and then write out the complete conjecture in $p$-adic case. We proved the theorem in good reduction and semistable reduction cases.
\r\n\r\nIn the second part of the paper, by using Berthelot, Esnault and R\\\"{u}lling's trace morphisms in [BER], we first prove the case of coherent cohomology, then we extend it to the Witt vector cohomology, and we then get a result on the Frobenius-stable part of the Witt vector cohomology, which corresponds the slope 0 part of the rigid cohomology, we then get the general $p$-adic local invariant cycle theorem.
\r\n\r\nWe also give another approach in the $H^0$ and $H^1$ cases in the general case.
\r\n\r\nIn the last part of the paper, based on Flach and Morin's work on the weight filtration in the $l$-adic case, we consider the $p$-adic analogous result (which, together with the $l$-adic's result, serves as a part to prove the compatibility of the Weil-etale cohomology with the Tamagawa number conjecture). This is a direct corollary of the local invariant cycle theorem by taking the weight filtration. And we also consider some typical examples that the weight filtration statement could be verified by direct computations.
", "doi": "10.7907/ZJ9C-WT04", "publication_date": "2012", "thesis_type": "phd", "thesis_year": "2012" }, { "id": "thesis:2017", "collection": "thesis", "collection_id": "2017", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05242006-225912", "primary_object_url": { "basename": "alpha.pdf", "content": "final", "filesize": 525637, "license": "other", "mime_type": "application/pdf", "url": "/2017/1/alpha.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "On the Equivariant Tamagawa Number Conjecture", "author": [ { "family_name": "Navilarekallu", "given_name": "Tejaswi", "clpid": "Navilarekallu-Tejaswi" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" }, { "family_name": "Oh", "given_name": "Hee", "orcid": "0000-0001-7978-7069", "clpid": "Oh-Hee" }, { "family_name": "Dimitrov", "given_name": "Mladen", "orcid": "0000-0002-7228-9136", "clpid": "Dimitrov-Mladen" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "For a finite Galois extension K/Q of number fields with Galois group G and a motive M = M' ⊗ h\u2070(Spec(K))(0) with coefficients in Q[G], the equivariant Tamagawa number conjecture relates the special value L*(M,0) of the motivic L-function to an element of K\u2080(Z[G];R) constructed via complexes associated to M. The conjecture for nonabelian groups G is very much unexplored. In this thesis, we will develop some techniques to verify the conjecture for Artin motives and motives attached to elliptic curves. In particular, we consider motives h\u2070(Spec(K))(0) for an A\u2084-extension K/Q and, h\u00b9 (E x Spec(L))(1) for an S\u2083-extension L/Q and an elliptic curve E/Q.\r\n", "doi": "10.7907/7HZ0-F068", "publication_date": "2006", "thesis_type": "phd", "thesis_year": "2006" }, { "id": "thesis:5020", "collection": "thesis", "collection_id": "5020", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-12162005-124435", "primary_object_url": { "basename": "thesis-final.pdf", "content": "final", "filesize": 507499, "license": "other", "mime_type": "application/pdf", "url": "/5020/1/thesis-final.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "On the Tamagawa Number Conjecture for Motives Attached to Modular Forms", "author": [ { "family_name": "Gealy", "given_name": "Matthew Thomas", "clpid": "Gealy-Matthew-Thomas" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Wambach", "given_name": "Eric", "clpid": "Wambach-E" }, { "family_name": "Oh", "given_name": "Hee", "clpid": "Oh-Hee" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We carry out certain automorphic and l-adic computations, the former extending results of Beilinson and Scholl, and the latter using ideas of Kato and Kings, related to explicit motivic cohomology classes on modular varieties. Under mild local and global conditions on a modular form, these give exactly the coordinates of the Deligne and l-adic realizations of said motivic cohomology class in the eigenspace attached to the modular form (Theorem 4.1.1). Assuming Kato's Main Conjecture and a Leopoldt-type conjecture, we deduce (a weak version of) the Tamagawa Number Conjecture for the motive attached to a modular form, twisted by a negative integer.", "doi": "10.7907/X671-G590", "publication_date": "2006", "thesis_type": "phd", "thesis_year": "2006" }, { "id": "thesis:2475", "collection": "thesis", "collection_id": "2475", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-06062005-134908", "type": "thesis", "title": "Artin L-Functions for Abelian Extensions of Imaginary Quadratic Fields", "author": [ { "family_name": "Johnson", "given_name": "Jennifer Michelle", "clpid": "Johnson-Jennifer-Michelle" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Wales", "given_name": "David B.", "clpid": "Wales-D-B" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "orcid": "0000-0002-0159-087X", "clpid": "Ramakrishnan-D" }, { "family_name": "Dimitrov", "given_name": "Mladen", "orcid": "0000-0002-7228-9136", "clpid": "Dimitrov-Mladen" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let F be an abelian extension of an imaginary quadratic field K with Galois group G. We form the Galois-equivariant L-function of the motive h(Spec F)(j) where the Tate twists j are negative integers. The leading term in the Taylor expansion at s=0 decomposes over the group algebra Q[G] into a product of Artin L-functions indexed by the characters of G. We construct a motivic element via the Eisenstein symbol and relate the L-value to periods via regulator maps. Working toward the equivariant Tamagawa number conjecture, we prove that the L-value gives a basis in etale cohomology which coincides with the basis given by the p-adic L-function according to the main conjecture of Iwasawa theory.", "doi": "10.7907/8T84-BQ83", "publication_date": "2005", "thesis_type": "phd", "thesis_year": "2005" }, { "id": "thesis:4595", "collection": "thesis", "collection_id": "4595", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-11182003-084742", "primary_object_url": { "basename": "BurnsFlachConjectureForIntegralHeckeAlgebra.pdf", "content": "final", "filesize": 417280, "license": "other", "mime_type": "application/pdf", "url": "/4595/1/BurnsFlachConjectureForIntegralHeckeAlgebra.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Bloch-Kato Conjecture for the Adjoint of H\u00b9(X\u2080(N)) with Integral Hecke Algebra", "author": [ { "family_name": "Lin", "given_name": "Qiang", "clpid": "Lin-Qiang" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Goins", "given_name": "Edray", "clpid": "Goins-E" }, { "family_name": "Aschbacher", "given_name": "Michael", "clpid": "Aschbacher-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Let M be a motive that is defined over a number field and admits an action of a finite dimensional semisimple Q-algebra T. David Burns and Matthias Flach formulated a conjecture, which depends on a choice of Z-order T in T, for the leading coefficient of the Taylor expansion at 0 of the T-equivariant L-function of M. For primes l outside a finite set we prove the l-primary part of this conjecture for the specific case where M is the trace zero part of the adjoint of H\u00b9(X\u2080(N)) for prime N and where T is the (commutative) integral Hecke algebra for cusp forms of weight 2 and the congruence group \u0393\u2080(N), thus providing one of the first nontrivial supporting examples for the conjecture in a geometric situation where T is not the maximal order of T.
\r\n \r\nWe also compare two Selmer groups, one of which appears in Bloch-Kato conjecture and the other a slight variant of what is defined by A. Wiles. A result on the Fontaine-Laffaille modules with coefficients in a local ring finite free over Z\u2113 is obtained.
", "doi": "10.7907/QD4X-J291", "publication_date": "2004", "thesis_type": "phd", "thesis_year": "2004" }, { "id": "thesis:1239", "collection": "thesis", "collection_id": "1239", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-04012004-151307", "primary_object_url": { "basename": "thesis.pdf", "content": "final", "filesize": 463793, "license": "other", "mime_type": "application/pdf", "url": "/1239/3/thesis.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "The Conjecture of Birch and Swinnerton-Dyer for Elliptic Curves with Complex Multiplication by a Nonmaximal Order", "author": [ { "family_name": "Colwell", "given_name": "Jason Andrew", "orcid": "0000-0002-1041-4739", "clpid": "Colwell-Jason-Andrew" } ], "thesis_advisor": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" } ], "thesis_committee": [ { "family_name": "Flach", "given_name": "Matthias", "clpid": "Flach-M" }, { "family_name": "Ramakrishnan", "given_name": "Dinakar", "clpid": "Ramakrishnan-D" }, { "family_name": "Goins", "given_name": "Edray", "clpid": "Goins-E" }, { "family_name": "Aschbacher", "given_name": "Michael", "clpid": "Aschbacher-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The Conjecture of Birch and Swinnerton-Dyer relates an analytic invariant of an elliptic curve -- the value of the L-function, to an algebraic invariant of the curve -- the order of the Tate--Shafarevich group. Gross has refined the Birch--Swinnerton-Dyer Conjecture in the case of an elliptic curve with complex multiplication by the full ring of integers in a quadratic imaginary field. It is this version which interests us here. Gross' Conjecture has been reformulated, by Fontaine and Perrin-Riou, in the language of derived categories and determinants of perfect complexes. Burns and Flach then realized that this immediately leads to a refined conjecture for elliptic curves with complex multiplication by a nonmaximal order. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. We prove this conjecture of Burns and Flach.", "doi": "10.7907/G40X-ST27", "publication_date": "2004", "thesis_type": "phd", "thesis_year": "2004" } ]