[ { "id": "authors:ahg63-7tj61", "collection": "authors", "collection_id": "ahg63-7tj61", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220802-839169000", "type": "article", "title": "Compatibility of Special Value Conjectures with the Functional Equation of Zeta Functions", "author": [ { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Morin", "given_name": "Baptiste", "clpid": "Morin-Baptiste" } ], "abstract": "We prove that the special value conjecture for the Zeta function \u03b6(X, s) of a proper, regular arithmetic scheme X that we formulated in [8] is compatible with the functional equation of \u03b6(X, s) provided that the rational factor C(X, n) we were not able to compute previously has the simple explicit form given in the introduction below.", "issn": "1431-0635", "publisher": "Deutsche Mathematiker-Vereinigung", "publication": "Documenta Mathematica", "publication_date": "2021", "volume": "26", "pages": "1633-1677" }, { "id": "authors:kg1dv-36j31", "collection": "authors", "collection_id": "kg1dv-36j31", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190607-083625026", "type": "article", "title": "Weil-\u00c9tale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes", "author": [ { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Baptiste", "given_name": "Morin", "clpid": "Baptiste-M" } ], "abstract": "We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme X at any integer n in terms of Weil-\u00e9tale cohomology complexes. This extends work of Lichtenbaum [65] and Geisser [36] for X of characteristic p, of Lichtenbaum [66] for X = Spec(O_F) and n = 0 where F is a number field, and of the second author for arbitrary X and n = 0 [72]. We show that our conjecture is compatible with the Tamagawa number conjecture of Bloch, Kato, Fontaine and Perrin-Riou [31] if X is smooth over Spec(O_F), and hence that it holds in cases where the Tamagawa number conjecture is known.", "doi": "10.25537/dm.2018v23.1425-1560", "issn": "1431-0635", "publisher": "Documenta Mathematica", "publication": "Documenta Mathematica", "publication_date": "2018", "volume": "23", "pages": "1425-1560" }, { "id": "authors:cp80m-x8671", "collection": "authors", "collection_id": "cp80m-x8671", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161027-120750559", "type": "article", "title": "On the local Tamagawa number conjecture for Tate motives over tamely ramified fields", "author": [ { "family_name": "Daigle", "given_name": "Jay", "clpid": "Daigle-J" }, { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" } ], "abstract": "The local Tamagawa number conjecture, which was first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic L-functions with the functional equation. The local conjecture was proven for Tate motives over finite unramified extensions K\u2215\u211a_p by Bloch and Kato. We use the theory of (\u03c6,\u0393)-modules and a reciprocity law due to Cherbonnier and Colmez to provide a new proof in the case of unramified extensions, and to prove the conjecture for \u211a_p(2) over certain tamely ramified extensions.", "doi": "10.2140/ant.2016.10.1221", "issn": "1937-0652", "publisher": "Mathematical Sciences Publishers", "publication": "Algebra and Number Theory", "publication_date": "2016-08-30", "series_number": "6", "volume": "10", "issue": "6", "pages": "1221-1275" }, { "id": "authors:5zd1m-jgn27", "collection": "authors", "collection_id": "5zd1m-jgn27", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150327-104226273", "type": "article", "title": "A note on the cohomology of the Langlands group", "author": [ { "family_name": "Fan", "given_name": "Edward S. T.", "clpid": "Fan-E-S-T" }, { "family_name": "Flach", "given_name": "M.", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" } ], "abstract": "We begin with a comparison of various cohomology theories for topological groups. Using the continuity result for Moore cohomology, we establish a Hochschild-Serre spectral sequence for a slightly larger class of groups. We use these properties to compute the cohomology of the Langlands group of a totally imaginary field. The appendix answers a question raised by Flach concerning the cohomological dimension of the group \u211d.", "doi": "10.1090/S0002-9947-2014-06230-2", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "2014-02-25", "series_number": "4", "volume": "367", "issue": "4", "pages": "2905-2920" }, { "id": "authors:pbr1f-jad54", "collection": "authors", "collection_id": "pbr1f-jad54", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120817-095043547", "type": "article", "title": "On the Weil-\u00c9tale Topos of Regular Arithmetic Schemes", "author": [ { "family_name": "Flach", "given_name": "M.", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Morin", "given_name": "B.", "clpid": "Moring-B" } ], "abstract": "We define and study a Weil-\u00e9tale topos for any regular,\nproper scheme X over Spec(Z) which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with \u02dcR-coefficients has the expected relation to \u03b6(X, s) at s = 0 if the Hasse-Weil L-functions L(h^(i)(X_(Q)), s) have the expected meromorphic\ncontinuation and functional equation. If X has characteristic p the cohomology with Z-coefficients also has the expected relation to \u03b6(X, s) and our cohomology groups recover those previously studied by Lichtenbaum and Geisser.", "issn": "1431-0635", "publisher": "Universit\u00e4t Bielefeld Fakultat Mathematik", "publication": "Documenta Mathematica", "publication_date": "2012", "volume": "17", "pages": "313-400" }, { "id": "authors:jhnm6-zqs39", "collection": "authors", "collection_id": "jhnm6-zqs39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120124-104112461", "type": "article", "title": "On the cyclotomic main conjecture for the prime 2", "author": [ { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" } ], "abstract": "We complete the proof of the equivariant Tamagawa number conjecture for Tate motives over absolutely abelian fields by proving a refined cyclotomic main conjecture at the prime 2.", "doi": "10.1515/CRELLE.2011.082", "issn": "0075-4102", "publisher": "Walter de Gruyter", "publication": "Journal F\u00fcr Die Reine und Angewandte Mathematik", "publication_date": "2011-12", "volume": "661", "pages": "1-36" }, { "id": "authors:f468v-awk92", "collection": "authors", "collection_id": "f468v-awk92", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:FLApamq09", "type": "article", "title": "Iwasawa Theory and Motivic L-functions", "author": [ { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" } ], "abstract": "We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture.", "issn": "1558-8602", "publisher": "International Press", "publication": "Pure and Applied Mathematics Quarterly", "publication_date": "2009-01", "series_number": "1, Sp.", "volume": "5", "issue": "1, Sp.", "pages": "255-294" }, { "id": "authors:sfth9-7qg70", "collection": "authors", "collection_id": "sfth9-7qg70", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180809-133558024", "type": "article", "title": "Cohomology of topological groups with applications to the Weil group", "author": [ { "family_name": "Flach", "given_name": "M.", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" } ], "abstract": "We establish various properties of the definition of cohomology of topological groups given by Grothendieck, Artin and Verdier in SGA4, including a Hochschild\u2013Serre spectral sequence and a continuity theorem for compact groups. We use these properties to compute the cohomology of the Weil group of a totally imaginary field, and of the Weil-\u00e9tale topology of a number ring recently introduced by Lichtenbaum (both with integer coefficients).", "doi": "10.1112/s0010437x07003338", "issn": "0010-437X", "publisher": "Oxford University Press", "publication": "Compositio Mathematica", "publication_date": "2008-05", "series_number": "03", "volume": "144", "issue": "03", "pages": "633-656" }, { "id": "authors:mmnvj-c4k12", "collection": "authors", "collection_id": "mmnvj-c4k12", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BURdm06", "type": "article", "title": "On the Equivariant Tamagawa Number Conjecture for Tate Motives, Part II", "author": [ { "family_name": "Burns", "given_name": "David", "clpid": "Burns-D" }, { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" } ], "abstract": "Let K be any finite abelian extension of Q, k any subfield of K and r any integer. We complete the proof of the equivariant Tamagawa Number Conjecture for the pair (h\u2070(Spec(K))(r),Z[Gal(K/k)]).", "issn": "1431-0635", "publisher": "Documenta Mathematica", "publication": "Documenta Mathematica", "publication_date": "2006", "volume": "Extra", "pages": "133-163" }, { "id": "authors:7a0sp-7at39", "collection": "authors", "collection_id": "7a0sp-7at39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BURdm01", "type": "article", "title": "Tamagawa Numbers for Motives with (Non-Commutative) Coefficients", "author": [ { "family_name": "Burns", "given_name": "D.", "clpid": "Burns-D" }, { "family_name": "Flach", "given_name": "M.", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" } ], "abstract": "Let $M$ be a motive which is defined over a number field and admits an action of a finite dimensional semisimple $\\bq$-algebra $A$. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at $0$ of the $A$-equivariant $L$-function of $M$. This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou et al. and also the central conjectures of classical Galois module theory as developed by Fr\u00f6hlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order $\\A$ in $A$ for which there exists a `projective $\\A$-structure' on $M$. The existence of such a structure is guaranteed if $\\A$ is a maximal order, and also occurs in many natural examples where $\\A$ is non-maximal. In each such case the conjecture with respect to a non-maximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in $A$ by making use of the category of virtual objects introduced by Deligne.", "issn": "1431-0635", "publisher": "Documenta Mathematica", "publication": "Documenta Mathematica", "publication_date": "2001", "volume": "6", "pages": "501-570" }, { "id": "authors:eaj7m-9zf33", "collection": "authors", "collection_id": "eaj7m-9zf33", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:FLAblms00", "type": "article", "title": "Euler characteristics in relative K-groups", "author": [ { "family_name": "Flach", "given_name": "M.", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" } ], "abstract": "Suppose that M is a finite module under the Galois group of a local or global field. Ever since Tate's papers [17, 18], we have had a simple and explicit formula for the Euler\u2013Poincar\u00e9 characteristic of the cohomology of M. In this note we are interested in a refinement of this formula when M also carries an action of some algebra [script A], commuting with the Galois action (see Proposition 5.2 and Theorem 5.1 below). This refinement naturally takes the shape of an identity in a relative K-group attached to [script A] (see Section 2). We shall deduce such an identity whenever we have a formula for the ordinary Euler characteristic, the key step in the proof being the representability of certain functors by perfect complexes (see Section 3). This representability may be of independent interest in other contexts. \n\nOur formula for the equivariant Euler characteristic over [script A] implies the 'isogeny invariance' of the equivariant conjectures on special values of the L-function put forward in [3], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer) was also a motivation for Tate's original paper [18]. I am very grateful to J-P. Serre for illuminating discussions on the subject of this note, in particular for suggesting that I consider representability. I should also like to thank D. Burns for insisting on a most general version of the results in this paper.", "doi": "10.1112/S0024609300006950", "issn": "0024-6093", "publisher": "London Mathematical Society", "publication": "Bulletin of the London Mathematical Society", "publication_date": "2000-05", "series_number": "3", "volume": "32", "issue": "3", "pages": "272-284" } ]