Information and Communication Sciences

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.

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