Curriculum Vitaes

Trihan Fabien Benoit

  (BENOIT TRIHAN FABIEN)

Profile Information

Affiliation
Associate Professor, Faculty of Science and Technology, Department of Information and Communication Sciences, Sophia University
Degree
学士(レンヌ第一大学)
修士(レンヌ第一大学)
DEA(レンヌ第一大学)
Ph.D(University of Rennes 1)
博士(純粋数学)(レンヌ第一大学)

Other name(s) (e.g. nickname)
Trihan Fabien
Researcher number
60738300
J-GLOBAL ID
201401054300838945
researchmap Member ID
7000007568

(Subject of research)
Geometric Iwasawa Theory


Research Interests

 1

Research Areas

 1

Papers

 23
  • King Fai Lai, Ignazio Longhi, Takashi Suzuki, Ki Seng Tan, Fabien Trihan
    Algebra and Number Theory, 15(4) 863-907, 2021  
    Let A be an abelian variety over a global function field K of characteristic p. We study the µ-invariant appearing in the Iwasawa theory of A over the unramified ℤp-extension of K. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate–Shafarevich group of A and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate–Shafarevich group (which is now the µ-invariant) in terms of other quantities including the Faltings height of A and Frobenius slopes of the numerator of the Hasse–Weil L-function of A/K assuming the conjectural Birch–Swinnerton-Dyer formula. Our next result is to prove this µ-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the “µ = 0” locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset.
  • Fabien Trihan, David Vauclair
    Proceedings of the American Mathematical Society, 149(9) 3601-3611, 2021  
    We state and prove certain cases of the equivariant Tamagawa number conjecture of a semistable Abelian variety over an everywhere unramified finite Galois extension of a global field of characteristic p > 0 under a semisimplicity hypothesis.
  • Trihan Fabien, Vauclair, David
    Documenta mathematica, 24 473-522, 2019  Peer-reviewed
    We establish the Iwasawa main onje ture for semistable abelian varieties over a function field of charateristi p under certain restrictive assumptions. Namely we consider p-torsion free p-adic Lie extensions of the base field which contain the constant Zp-extension and are everywhere unramifield. Under the usual μ = 0 hypothesis, we give a proof which mainly relies on the interpretation of the Selmer complex in terms of p-adic cohomology [TV] together with the trace formulas of [EL1].
  • King Fai Lai, Ignazio Longhi, Ki-Seng Tan, Fabien Trihan
    Transactions of the American Mathematical Society, 370(3) 1925-1958, 2018  Peer-reviewed
    We prove a functional equation for two projective systems of finite abelian p-groups, {an} and {abn}, endowed with an action of ℤdp such that an can be identified with the Pontryagin dual of bn for all n. Let K be a global field. Let L be a ℤdp-extension of K (d ≥ 1), unramified outside a finite set of places. Let A be an abelian variety over K. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of A.
  • King Fai Lai, Ignazio Longhi, Ki-Seng Tan, Fabien Trihan
    PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 112(6) 1040-1058, Jun, 2016  
    We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over Z(p)(d)-extensions of function fields ramifying at a finite set of places.
  • King Fai Lai, Ignazio Longhi, Ki-Seng Tan, Fabien Trihan
    MATHEMATISCHE ZEITSCHRIFT, 282(1-2) 485-510, Feb, 2016  
    We prove the Iwasawa Main Conjecture over the arithmetic -extension for semistable abelian varieties over function fields of characteristic .
  • King Fai Lai, Ignazio Longhi, Ki-Seng Tan, Fabien Trihan
    Proceedings of the American Mathematical Society, 143(6) 2355-2364, 2015  
    Let A/K be an elliptic curve over a global field of characteristic p &gt 0. We provide an example where the Pontrjagin dual of the Selmer group of A over a Γ := ℤp-extension L/K is not a torsion ℤp[[Γ]]-module and show that the Iwasawa Main Conjecture for A/L holds nevertheless.
  • Fabien Trihan, Seidai Yasuda
    Compositio Mathematica, 150(4) 507-522, 2014  
    Let A/K be an abelian variety over a function field of characteristic p &gt 0 and let l be a prime number (l = p allowed). We prove the following: the parity of the corank rl of the l-discrete Selmer group of A/K coincides with the parity of the order at s = 1 of the Hasse-Weil L-function of A/K. We also prove the analogous parity result for pure l-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin-Tate conjecture. © The Author(s) 2014.
  • Fabien Trihan, Christian Wuthrich
    COMPOSITIO MATHEMATICA, 147(4) 1105-1128, Jul, 2011  
    We prove the p-parity conjecture for elliptic curves over global fields of characteristic p > 3. We also present partial results on the l-parity conjecture for primes l not equal p.
  • TADASHI OCHIAI, FABIEN TRIHAN
    Mathematical Proceedings of the Cambridge Philosophical Society, 146(1) 23-43, Jan, 2009  
    <title>Abstract</title>We study a (<italic>p</italic>-adic) geometric analogue for abelian varieties over a function field of characteristic <italic>p</italic> of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.
  • Olivier Brinon, Fabien Trihan
    Rendiconti del Seminario Matematico della Università di Padova, 119 141-171, 2008  
  • Fabien Trihan
    JOURNAL OF MATHEMATICAL SCIENCES-THE UNIVERSITY OF TOKYO, 15(3) 411-425, 2008  Peer-reviewed
    We show that the Dieudonne crystal associated to a Barsotti-Tate group with potentially semistable reduction over a smooth curve is overconvergent. As a corollary, we obtain the rationality of the L-function associated to this group.
  • Christine Noot-Huyghe, Fabien Trihan
    Annales de la Faculté des sciences de Toulouse : Mathématiques, 16(3) 611-634, 2007  Peer-reviewed
  • Shigeki Matsuda, Fabien Trihan
    Journal für die reine und angewandte Mathematik (Crelles Journal), 2004(569) 47-54, Jan 30, 2004  
    Let X be a proper smooth curve over a perfect field of characteristic p > 0 and U an open dense subscheme of X. We prove that convergent F-isocrystals on U are overconvergent under the condition that they are overconvergent at each point in X/U. Using this criterion, we show that the higher direct images Rifcrys* script O sign V by a proper smooth morphism of schemes f : V → U are overconvergent.
  • Kazuya Kato, Fabien Trihan
    Inventiones Mathematicae, 153(3) 537-592, Sep 1, 2003  
  • Fabien Trihan
    J. Math. Sci. Univ. Tokyo, 9(2) 279-301, 2002  Peer-reviewedLead author
  • Fabien Trihan
    Rend. Sem. Mat. Univ. Padova, 108 1-26, 2002  Peer-reviewedLead author
    We generalize the study of the syntomic cohomology of [17] to the case of open varieties and with coefficients.
  • Fabien Trihan
    Comptes Rendus de l'Academie des Sciences - Series I: Mathematics, 332(5) 431-436, Mar 1, 2001  
    We give an expression of the zeroes and poles of the L-function of an overconvergent F-isocrystal defined on an open variety when the geometric situation lifts to the characteristic 0 and under the hypothesis that the connexion of the isocrystal is regular at infinity and satisfies a condition on the residues. © 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.
  • F Trihan
    COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE, 332(5) 431-436, Mar, 2001  Peer-reviewed
    We give m? expression of the zeroes and poles of the L-function of an overconvergent F-isocrystal defined on an open variety when the geometric situation lifts to the characteristic 0 and under the hypothesis that the connexion of the isocrystal is regular at infinity and satisfies a condition on the residues. (C) 2001 Academie des sciences/Editions scientifiques ct medicales Elsevier SAS.
  • Le Stum, B, Trihan, F
    Ann. Inst. Fourier (Grenoble), 51(5) 1189-1207, 2001  Peer-reviewedLead author
  • F Trihan
    COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE, 331(3) 229-234, Aug, 2000  Peer-reviewed
    Let X a proper and smooth scheme over a finite field, D a divisor with normal crossings and U the complementary of D in X. We give a description of the zeroes and poles of the Zeta function of these three schemes, when they lift to the characteristic 0. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.
  • Fabien Trihan
    Manuscripta Mathematica, 96(4) 397-419, Aug, 1998  
    The purpose of this article is to give a cohomological formula for the unit-root part of the L-function associated to a Barsotti-Tate group G on a scheme S over a field of characteristic p when G extends to some compactification of S. This is an analogue of a part of a conjecture of Katz according to wich the L-function of an F-crystal should be expressed in terms of the p-adic etale sheaf corresponding to the unit-root part of the crystal. In order to carry out this project, we use the technics of [E-LS II] wich require in our case an extension of the Dieudonné crystalline theory ([B-B-M]) to "crystal of level m" in the sense of Berthelot. We show that the unit-root L-function of the Dieudonné crystal associated to G can be expressed in terms of the syntomic cohomology of the Ext group of G by the constant sheaf.
  • F Trihan
    MANUSCRIPTA MATHEMATICA, 96(4) 397-419, Aug, 1998  Peer-reviewed
    The purpose of this article is to give a cohomological formula for the unit-root part of the L-function associated to a Barsotti-Tate group G on a scheme S over a field of characteristic p when G extends to some compactification of S. This is an analogue of a part of a conjecture of Katz according to wich the L-function of an F-crystal should be expressed in terms of the p-adic etale sheaf corresponding to the unit-root part of the crystal. In order to carry out this project, we use the technics of [E-LS II] wich require in our case an extension of the Dieudonne crystalline theory ([B-B-M]) to "crystal of level m" in the sense of Berthelot. We show that the unit-root L-function of the Dieudonne crystal associated to G can be expressed in terms of the syntomic cohomology of the Err group of G by the constant sheaf.

Books and Other Publications

 2
  • Trihan Fabien Benoit (Role: Joint editor, p.1-337)
    Birkhäuser , Springer, Dec 4, 2014 (ISBN: 9783034808521)
    This volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009-2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields.
  • Trihan Fabien Benoit (Role: Joint author, p.119-181)
    Birkhäuser , Springer, Dec 4, 2014 (ISBN: 9783034808521)
    We give a proof of the Iwasawa Main Conjecture for smooth Zp-sheaves (resp. semistable abelian varieties) over (resp. unramified) p-adic Lie extensions of function fields

Research Projects

 2