Research - Ivan Fesenko

List of sections:

R Recent
M Epidemic modelling
L Anabelian geometry, IUT theory, and applications
K 2d adelic analysis and geometry, and applications
J Adelic structures on arithmetic and geometric surfaces, and applications
I Higher integration, harmonic analysis and zeta integrals
H Interactions of model theory, arithmetic and algebraic geometry and noncommutative geometry
G Arithmetic noncommutative class field theory and local reciprocity maps
F Infinite ramification theory and pro-p-group theory
E Local field arithmetic, representation theory of algebraic groups
D Ramification theory, finite and infinite Galois extensions, perfect and imperfect residue field
C Higher local fields, their structures, their algebraic K-groups
B Class field theories, one-dimensional and higher dimensional
A Explicit formulas for generalized Hilbert symbol on local and higher local fields
Textbooks
Surveys
Volumes edited

some files are updated versions of the published versions

R Recent

[R15] From two Grothendieck’s math legacies to quantum computing and deep neural networks, talk July 2024, KIAS, Seoul
[R14] The effective abc inequality and how it was applied to a new proof of FLT, Zhejiang university, talk October 2023
[R13] On new interactions between quantum theories and arithmetic geometry, October 2023
[R12] Core topics in number theory I (algebraic number theory, class field theory)
[R11] Problems in higher adelic theory, talk April 2023 Beijing
[R10] Anabelian geometry-IUT-effective abc inequalities-applications, Tsinghua university, talk March 2023
[R9] Categories, toposes, anabelian geometry, IUT and quantum computing, 110 minutes talk, Crypto-Math CREST workshop, Tokyo, September 2022
[R8] Sh. Mochizuki, I. Fesenko, Yu. Hoshi, A. Minamide, W. Porowski, Explicit estimates in inter-universal Teichmüller theory, Kodai Math. J. 45(2022) 175-236
[R7] Geometries underlying deep properties of numbers, video of a talk at Institute of Mathematics, Kyiv, Ukraine
[R6] Higher adelic theory, video of a talk at the Como School, September 2021
[R5] IUT and modern number theory, talk at the RIMS workshop on IUT Summit, September 2021
[R4] Class field theory, its three main generalisations, and applications, EMS Surveys 8(2021) 107-133
[R3] A. Zhigljavsky, J. Noonan, I. Fesenko, K. Kremnitzer et al, A prototype decision support tool for handling the COVID-19 UK epidemic, April 2020
[R2] A. Zhigljavsky, I. Fesenko, et al, Comparison of different exit scenarios from the lock-down for COVID-19 epidemic in the UK and assessing uncertainty of the predictions, April 2020
[R1] A. Zhigljavsky, I. Fesenko et al, Generic probabilistic modelling and non-homogeneity issues for the UK epidemic of COVID-19, April 2020

M Epidemic modelling

[M3] A. Zhigljavsky, J. Noonan, I. Fesenko, K. Kremnitzer et al, A prototype decision support tool for handling the COVID-19 UK epidemic, April 2020
[M2] A. Zhigljavsky, I. Fesenko, et al, Comparison of different exit scenarios from the lock-down for COVID-19 epidemic in the UK and assessing uncertainty of the predictions, April 2020
[M1] A. Zhigljavsky, I. Fesenko et al, Generic probabilistic modelling and non-homogeneity issues for the UK epidemic of COVID-19, April 2020

L Anabelian geometry, IUT theory, and applications

[L5] On new interactions between quantum theories and arithmetic geometry, October 2023
[L4] Sh. Mochizuki, I. Fesenko, Yu. Hoshi, A. Minamide, W. Porowski, Explicit estimates in inter-universal Teichmüller theory, Kodai Math. J. 45(2022) 175-236
[L3] Class field theory, its three main generalisations, and applications, EMS Surveys 8(2021) 107-133
[L2] Fukugen, Inference: International Review of Science 2 no. 3 (2016)
[L1] Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Europ. J. Math. (2015) 1:405–440

The effective abc inequality and how it was applied to a new proof of FLT, Zhejiang university, talk October 2023
Anabelian geometry-IUT-effective abc inequalities-applications, Tsinghua university, talk March 2023
Categories, toposes, anabelian geometry, IUT and quantum computing, Crypto-Math CREST workshop talk, Tokyo, September 2022
Introducing anabelian geometry, a general talk, 2021
Geometries underlying deep properties of numbers, video of a talk at Institute of Mathematics, Kyiv, Ukraine 2021
IUT and modern number theory, talk at RIMS workshop on IUT Summit, September 2021
On inter-universal Teichmüller theory of Shinichi Mochizuki, generic 90 minutes talk
Reciprocity and IUT, talk at RIMS/S&C workshop on IUT Summit, Kyoto 2016
Extended proceedings of Oxford 2015 IUT workshop
Guides on IUT

K 2d adelic analysis and geometry, and applications

[K6] Selective integration on higher adeles and the Euler characteristic for surfaces, work in progress
[K5] Analysis on arithmetic schemes, III, work in progress
[K4] Class field theory, its three main generalisations, and applications, EMS Surveys 8(2021) 107-133
[K3] I. Fesenko, G. Ricotta, M. Suzuki, Mean-periodicity and zeta functions, Ann. L’Inst. Fourier, 62 (2012) 1819-188
[K2] Analysis on arithmetic schemes. II, J. K-theory 5 (2010) 437-557
[K1] Adelic approach to the zeta function of arithmetic schemes in dimension two, Moscow Math. J. 8 (2008) 273-317

Problems in higher adelic theory, talk
Remaining open problems in 2d adelic analysis and geometry
Higher adelic theory, video of a 75 minutes talk
Adelic geometry and analysis on regular models of elliptic curves over global fields and their zeta functions, 90 minutes talk

For related developments see
W. Czerniawska, Higher adelic programme, adelic Riemann-Roch theorems and the BSD conjecture, PhD thesis, eprints Nottingham 2018
T. Oliver, Zeta integrals on arithmetic surfaces, St. Petersburg Math. J. 27(2016)

For some of related analytic aspects of zeta functions see

M. Suzuki, Two-dimensional adelic analysis and cuspidal automorphic representations of GL(2), 339-361, In Multiple Dirichlet Series, L-functions and automorphic forms, eds., Progress in Math. 300, Birkhauser 2012
M. Suzuki, Positivity of certain functions associated with analysis on elliptic surfaces, J. Number Theory 131 (2011) 1770-1796
M. Suzuki, On zeta integrals related to Hasse-Weil L-functions of elliptic curves, RIMS, Kokyuroku 1665 (2009) 105-113
T. Oliver, Hecke characters and the mean-periodicity correspondence for CM elliptic curves, arXiv:1307.6706
T. Oliver, Mean-periodicity and automorphicity, J. Math. Soc. Japan 69 (2017) 25-51

J Adelic structures on arithmetic and geometric surfaces, and applications
[J3] Geometric adeles and the Riemann–Roch theorem for 1-cycles on surfaces, Moscow Math. J. 15 (2015) 435-453
[J2] Analysis on arithmetic schemes. II, J. K-theory 5 (2010) 437-557
[J1] Adelic approach to the zeta function of arithmetic schemes in dimension two, Moscow Math. J. 8 (2008) 273-317

Higher adelic theory, video of a 75 minutes talk

For some of related arithmetic and geometric aspects see

W. Czerniawska, P. Dolce, Adelic geometry on arithmetic surfaces II: completed adeles and idelic Arakelov intersection theory, J. Number Theory 2019, arxiv: 1906:03745
P. Dolce, Adelic geometry on arithmetic surfaces I: idelic and adelic interpretation of Deligne pairing, arxiv: 1812.10834
P. Dolce, Low dimensional adelic geometry, PhD thesis, eprints Nottingham 2018
M. Morrow, Grothendieckʼs trace map for arithmetic surfaces via residues and higher adeles, Algebra & Number Th., 2012, 6-7 (2012) 1503-1536
M. Morrow, An explicit approach to residues on and dualizing sheaves of arithmetic surfaces, New York J. Math., 16 (2010) 575-627
O. Bräunling, Adele residue symbol and Tate’s central extension for multiloop Lie algebras, arXiv:1206.2025
O. Bräunling, Two-dimensional ideles with cycle module coefficients, arXiv:1101.0424

I Higher integration, harmonic analysis and zeta integrals
[I2] Measure, integration and elements of harmonic analysis on generalized loop spaces, Proceed. St. Petersburg Math. Soc., vol. 12 (2005), 179-199; AMS Transl. Series 2, vol. 219, 149-164, 2006
[I1] Analysis on arithmetic schemes. I, Docum. Math.(2003) 261-284

For related measure and integration aspects see
M. Waller, Measure and integration on GL2 over a two-dimensional local field, arXiv:1902.02899, New J. Math. 25 (2019) 396-422
M. Waller, An approach to harmonic analysis on non-locally compact groups I: level structures over locally compact groups, arXiv:1902.02909
M. Waller, An approach to harmonic analysis on non-locally compact groups II: an invariant measure on groups of ordered type, arXiv:1902.02913
M. Morrow, Integration on valuation fields over local fields, Tokyo J. Math., 33 (2010) 235-281
M. Morrow, Integration on product spaces and GL_n of a valuation field over a local field, Comm. in Number Th. and Physics, 2 (2008) 563-592
M. Morrow, Fubiniʼs theorem and non-linear changes of variables over a two-dimensional local ﬁeld, arXiv:0712.2177

H Interactions of model theory, arithmetic and algebraic geometry and noncommutative geometry
[H3] Model theory guidance in number theory? – In Model Theory with Applications to Algebra and Analysis, LMS Lecture Note Series, 349, CUP, 2008, 327-334
[H2] Several nonstandard remarks, – In AMS/IP Advances in the Mathematical Sciences, AMS Transl. Series 2, vol. 217 (2006) 37-50
[H1] Remark 1 sect. 4 and Remark 3 sect. 13 of Analysis on arithmetic schemes. I, Docum. Math., (2003) 261-284

G Arithmetic noncommutative class field theory and local reciprocity maps
[G3] On the image of noncommutative reciprocity map, Homology, Homotopy and Applications, 7 (2005) 53-62
[G2] Noncommutative (nonabelian) local reciprocity maps, In Class Field Theory – Its Centenary and Prospects, Advanced Studies in Pure Math., vol. 30, 63-78, Math. Soc. Japan, Tokyo 2001
[G1] Local reciprocity cycles, in Invitation to higher local fields, I. Fesenko, M. Kurihara (eds.) , Geometry and Topology Monographs, Warwick 2000, pp. 293-298.

see also
K.I. Ikeda, E. Serbest, Fesenko reciprocity map, St. Petersburg Math. J. 20 (2008) 407-445
K.I. Ikeda, E. Serbest, Generalized Fesenko reciprocity map, St. Petersburg Math. J. 20 (2008) 593-624
K.I. Ikeda, E. Serbest, Non-abelian local reciprocity law, Manuscr. Math. 132 (2010) 19-49

F Infinite ramification theory and pro-p-group theory
[F2] M. du Sautoy, I. Fesenko, Where the wild things are: ramification groups and the Nottingham group, In New horizons in pro-p groups, 287-328, Progr. Math., 184, Birkhauser 2000
[F1] On just infinite pro-p-groups and arithmetically profinite extensions of local fields, J. Reine Angew. Mathematik 517 (1999) 61-80

E Local field arithmetic, representation theory of algebraic groups
[E2] Last section of Adelic approach to the zeta function of arithmetic schemes in dimension two, Moscow Math J 8 (2008) 273-317
[E1] I. Efrat and I. Fesenko, Fields Galois-equivalent to a local field of positive characteristic, Math. Res. Lett. 6 (1999) 345-356

D Ramification theory, finite and infinite Galois extensions, perfect and imperfect residue field
[D4] Ch. 3 of I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions, Second extended edition, AMS 2002, 341 pp.
[D3] On deeply ramiﬁed extensions, Journal of the LMS (2) 57 (1998) 325-335
[D2] Hasse-Arf property and abelian extensions, Math. Nachr. 174 (1995) 81-87
[D1] Abelian local p-class ﬁeld theory, Math. Ann. 301 (1995) 561-586.

C Higher local fields, their structures, their algebraic K-groups
[C5] I. B. Fesenko, S. V. Vostokov, S. H. Yoon, Generalised Kawada-Satake method for Mackey functors in class field theory, Europ. J. Math. 4(2018) 953-987
[C4] Ch. 9 of I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions, Second extended edition , AMS 2002
[C3] Sequential topologies and quotients of Milnor K-groups of higher local fields, with appendix by O.T. Izhboldin, Algebra i Analiz, 13 (2001), issue 3, 198-228; St. Petersburg Math. J. 13 (2002) 485-501
[C2] Topological Milnor K-groups of higher local ﬁelds, in Invitation to higher local fields, I. Fesenko, M. Kurihara (eds.), Geometry and Topology Monographs, Warwick 2000, pp. 61-74
[C1] On K-groups of a multidimensional local ﬁeld, Ukraine Mat. J. 41 issue 2 (1989), 266-268; English transl. in Ukrainian Math. J. 41(1989) 237-240

see also
A. Camara, Interaction of topology and algebra in arithmetic geometry, PhD thesis 2013

B Class field theories, one-dimensional and higher dimensional
[B17] Core topics in number theory I (algebraic number theory, class field theory)
[B16] Class field theory, its three main generalisations, and applications, May 2021, EMS Surveys 8(2021) 107-133
[B15] I. B. Fesenko, S. V. Vostokov, S. H. Yoon, Generalised Kawada-Satake method for Mackey functors in class field theory, Europ. J. Math. 4(2018) 953-987
[B14] Chapter 3 of Analysis on arithmetic schemes. II, J. K-theory 5 (2010) 437-557
[B13] Ch. 4, 5 of I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions, Second extended edition , AMS 2002, 341 pp.
[B12] Explicit higher local class ﬁeld theory, in Invitation to higher local fields, I. Fesenko, M. Kurihara (eds.), Geometry and Topology Monographs, Warwick 2000, pp. 95-101
[B11] Higher class ﬁeld theory without using K-groups, in Invitation to higher local fields, I. Fesenko, M. Kurihara (eds.), Geometry and Topology Monographs, Warwick 2000, pp. 137-142
[B10] Complete discrete valuation ﬁelds. Abelian local class ﬁeld theories, in Handbook of Algebra (man. ed. M. Hazewinkel), vol. 1, pp. 221-268, Elsevier, Amsterdam 1996
[B9] On general local reciprocity maps, J. reine angew. Math. 473 (1996) 207-222
[B8] Abelian extensions of complete discrete valuation ﬁelds, Number Theory Paris 1993/94, Cambridge Univ. Press 1996, 47-74
[B7] Abelian local p-class ﬁeld theory, Math. Ann. 301 (1995) 561-586
[B6] Local class ﬁeld theory: perfect residue ﬁeld case, Izvest. Russ. Acad. Nauk. Ser. Mat. 57 issue 4(1993) 72-91; English transl. in Russ. Acad. Sc. Izv. Math. 43 (1994) 65-81
[B5] On norm subgroups of complete discrete valuation ﬁelds, Vestn. St. Petersburg Univ. Series I, issue 2 1993, 54-57
[B4] Class ﬁeld theory of multidimensional local ﬁelds of characteristic 0, with the residue ﬁeld of positive characteristic, Algebra i Analiz 3 issue 3 (1991) 165-196; English transl. in St. Petersburg Math. J. 3 (1992) 649-678
[B3] Multidimensional local class ﬁeld theory.II, Algebra i Analiz 3 issue 5 (1991) 168-189; English transl. in St. Petersburg Math. J. 3 (1992) 1103-1126
[B2] Multidimensional local class ﬁeld theory, Dokl. AN SSSR 318 issue 1 (1991) 47-50; English transl. in Acad. Scienc. Dokl. Math. 43 (1991) 674-677
[B1] On class ﬁeld theory of multidimensional local ﬁelds of positive characteristic, Adv. Sov. Math. 4 (1991) 103-127.

see also
K. Syder, Reciprocity laws for the higher tame symbol and the Witt symbol on an algebraic surface, arXiv:1304.6250
K. Syder, Two-Dimensional local-global class field theory in positive characteristic, arXiv:1403.6747

A Explicit formulas for generalized Hilbert symbol on local and higher local fields
[A7] Ch. 7,8 of I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions, Second extended edition , AMS 2002
[A6] The generalized Hilbert symbol in multidimensional local ﬁelds, in Rings and Modules, vyp. 2, 1988, 88-92
[A5] S.V. Vostokov, I.B. Fesenko, A property of the Hilbert pairing, Matem. Zametki 43(1988), 393-400; English transl. in Mathem. Notes 43 (1988) 226-230
[A4] Explicit formulas for the generalized Hilbert symbol on Lubin-Tate formal groups – see Ch. I of 1987 thesis part i part ii part iii
[A3] The generalized Hilbert symbol in 2-adic case, Vestnik St. Petersburg Univ. 1985 issue 22, 112-114; English transl. in Vestnik St Petersburg Univ. Math. 18 (1985) 88-91
[A2] The Hilbert symbol on Lubin-Tate formal groups. III, in Rings and matrix groups, 1984, 146-150
[A1] S.V. Vostokov, I.B. Fesenko, The Hilbert symbol on Lubin-Tate formal groups. II, Zapiski nauch. semin. LOMI 132(1983) 85-96; English transl. in J. Soviet Math. 30 (1985)

Textbooks
[T4] Core topics in number theory I (algebraic number theory, class field theory)
[T3] Invitation to higher local fields, I. Fesenko, M. Kurihara (eds.) Geometry and Topology Monographs vol 3, Warwick 2000, part I, part II
[T2] I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions, Second extended edition , AMS 2002, 341 pp.
[T1] I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions, A Constructive Approach, AMS 1993, 284 pp.

Surveys
[S7] On new interactions between quantum theories and arithmetic geometry, and beyond
[S6] Fukugen, Inference: International Review of Science 2 no. 3 (2016)
[S5] Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Europ. J. Math. (2015) 1:405–440
[S4] Adelic approach to the zeta function of arithmetic schemes in dimension two, Moscow Math. J. 8 (2008), 273-317
[S3] I. Fesenko, M. du Sautoy, Where the wild things are: ramiﬁcation groups and the Nottingham group, in New horizons in pro-p-groups, Birkhaeuser, 2000, 287-328
[S2] Abelian extensions of complete discrete valuation ﬁelds, Number Theory Paris 1993/94, Cambridge Univ. Press 1996, 47-74
[S1] Complete discrete valuation ﬁelds. Abelian local class ﬁeld theories, in Handbook of Algebra (man. ed. M. Hazewinkel), vol. 1, pp. 221-268, Elsevier, Amsterdam 1996

Volumes edited
[V7] EMS Surveys Volume of 2018 Fuerteventura workshop, 2021, F. Bogomolov, I. Cheltsov, I. Fesenko (eds.)
[V6] Documenta Mathematica Volume dedicated to A.S. Merkuriev, 2015, P. Balmer, V. Chernousov, I. Fesenko, E. Friedlander, S. Garibaldi, Z. Reichstein, U. Rehmann (eds.)
[V5] Documenta Mathematica Volume dedicated to A.A. Suslin, 2010, 723 pp., I. Fesenko, E. Friedlander, A. Merkuriev, U. Rehmann (eds.)
[V4] Documenta Mathematica Volume dedicated to J.H. Coates, 2006, 826 pp., I. Fesenko, S. Lichtenbaum, B. Perrin-Riou, P. Schneider (eds.)
[V3] Volumes 11 and 12 of St Petersburg Mathematical Society Proceedings dedicated to S.V. Vostokov, 2005, 416 pp.; English translation by the AMS, Transl. Series 2 vol. 218, 219, 2006: Volume 1 Volume 2; I. Fesenko, I. Zhukov (eds.)
[V2] Documenta Mathematica Volume dedicated to K. Kato, 2003, 918 pp., S. Bloch, I. Fesenko, L. Illusie, M. Kurihara, S. Saito, T. Saito, P. Schneider (eds.)
[V1] Invitation to higher local fields, I. Fesenko, M. Kurihara (eds.), Geometry and Topology Monographs vol 3, Warwick 2000, part I part II