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General

My mathematical research interest lies in algebraic number theory. I am now working on Iwasawa theory, which is regarded as the best way to understand the arithmetic meaning of zeta and L-values of various motives. Especially, Iwasawa main conjectures give the precise connection between the congruence properties of the special values of L-functions (p-adic L-functions) and the (dual of p-primary part of) Selmer groups of a given motive. Many interesting arithmetic properties are encoded in Selmer groups; for example, in the case of elliptic curves, Selmer groups contain the information of the rational points of elliptic curves (Mordell-Weil groups) and the failure of the local-global principle (Shafarevich-Tate groups).

Research Themes

  • Anticyclotomic Iwasawa theory for modular forms à la Bertolini-Darmon, Gross, and Vatsal.
  • Overconvergent construction of (anticyclotomic) p-adic L-functions.
  • A bit more explicit computation of Bloch-Kato's exponential and dual exponential maps.
  • Applications of modularity lifting theorems to Iwasawa theory
  • Refined Iwasawa theory
  • Iwasawa theory for elliptic curves at additive reduction
  • Hida and Coleman families
  • p-adic variation of Euler and Kolyvagin systems.
  • Kato's epsilon conjecture.
  • Applications of machine learning (any ``applied mathematics") to number theory (Hope I can do this soon..)

    Math Writings

  • Remarks on Kato's Euler systems for elliptic curves with additive reduction, with Kentaro Nakamura
          submitted
  • Indivisibility of Heegner points and arithmetic applications, with Ashay Burungale and Francesc Castella
          submitted
  • On the refined conjectures on Fitting ideals of Selmer groups of elliptic curves with supersingular reduction, with Masato Kurihara
          submitted
  • On the indivisibility of derived Kato's Euler systems and the main conjecture for modular forms, with Myoungil Kim, Hae-Sang Sun
          submitted
  • An anticyclotomic Mazur-Tate conjecture for modular forms
          submitted
  • Overconvergernt quaternionic forms and anticyclotomic p-adic L-functions
          Publicacions Matem\`{a}tiques, to appear
  • Variation of anticyclotomic Iwasawa invariants in Hida families (arXiv), with Francesc Castella and Matteo Longo
          Algebra \& Number Theory, December 2017, Vol. 11, No. 10, pp. 2339--2368
  • On the freeness of anticyclotomic Selmer groups of modular forms (pdf), with Robert Pollack and Tom Weston
          International Journal of Number Theory, July 2017, Vol. 13, No. 06, pp. 1443--1455
  • Anticyclotomic Iwasawa invariants and congruences of modular forms (pdf) (This is my thesis.)
          Asian Journal of Mathematics, June 2017, Vol. 21, No. 3, pp. 499--530

    In preparation (to push myself..)

  • (title undecided), with Kazuto Ota
          in preparation
  • (title undecided), with Jaehoon Lee
          in preparation
  • (title undecided), with Jaehoon Lee and Gautier Ponsinet
          in progress

    Interesting math and non-math stuff on the internet

  • Thinking Space
  • How to study spectral sequences
  • Geometric vs. Arithmetic Frobenii
  • The topology of the Robba ring
  • BSD, p-adic BSD, and Iwasawa main conjecture
  • Flatness in algebraic geometry
  • Almost purity theorem
  • Vanishing cycles
  • Computing hypercohomology
  • Monodromy
  • Computing monodromy
  • Monodromy and global cohomology
  • Pontryagin dual and mu invariants
  • Ring homomorphisms
  • *Ordinary* Galois representations
  • Ordinary local Galois invariants
  • Well-definedness of modular Jacobians
  • On the module structure of local systems
  • Characteristic complexes in Iwasawa theory
  • The difference between an etale finite group scheme and a finite group
  • Why does the Section Conjecture exclude curves of genus 1?
  • Prime 2 and 3
  • Prime 2
  • Various topologies in algebraic geometry
  • Flat topology
  • Local properties of schemes
  • Mayer-Vietoris
  • Timeline of class field theory
  • Finding the Zariski closure of a set
  • Around locally ringed spaces. See also *Enlightning Exercise 4.3.A* in Vakil's book
  • Intuition for etale morphisms
  • 5/8 bound in group theory
  • Are there Maass forms where the expected Galois representation is l-adic?
  • Reference book for Galois Representations
  • Frobenius weights on etale cohomology and purity
  • Semisimplicity of Frobenius on *integral* Tate module
  • Psi operator on Phi-Gamma modules
  • Refereeing a Paper
  • Demonstrating that rigour is important
  • What is the motivation for a vertex algebra?
  • Heuristic argument that finite simple groups _ought_ to be "classifiable"?
  • Reading the mind of Prof. John Coates (motive behind his statement)
  • etale cohomology of an abelian variety and its dual
  • Why do we care about dual spaces?
  • Counter-examples for the quasi-unipotence of monodromy over an annulus?
  • Torsors in Algebraic Geometry?
  • What does the Lefschetz principle (in algebraic geometry) mean exactly?
  • Avoiding Minkowski's theorem in algebraic number theory
  • Do Tamagawa numbers of Galois representations stabilise in the cyclotomic tower?
  • How to avoid any wrong elementary ``proofs" of Fermat's last theorem