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Ivan Fesenko - Research Texts


some papers are included several times in different sections, depending on which of their aspects are relevant 

several files are updated versions of the published versions



L Two-dimensional theories in arithmetic geometry 

- Anabelian geometry and Sh. Mochizuki's inter-universal Teichmueller theory


[50] Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Europ. J. Math., 2015, doi: 10.1007/s40879-015-0066-0 pdf  


K Zeta functions and higher adelic zeta integrals 

- Meromorphic continuation of zeta functions of schemes of finite type and mean-periodicity correspondence

- Positivity hypothesis and the GRH  for zeta functions of regular models of elliptic curves

- Interaction between two adelic structures on surfaces and  the zeta function at central point


[49] 2d zeta integral, geometry of 2d adeles and the BSD conjecture for elliptic curves over global fields (Analysis on arithmetic schemes. III), work in progress  pdf 

[48] I. Fesenko, G. Ricotta, M. Suzuki, Mean-periodicity and zeta functions, Ann. L'Inst. Fourier, 62 (2012), 1819-1887 pdf  

[47] Analysis on arithmetic schemes. II, J. K-theory 5 (2010), 437-557 pdf  

[46] Adelic approach to the zeta function of arithmetic schemes in dimension two, Moscow Math. J. 8 (2008), 273-317 pdf  


40 research problems on adelic analysis and geometry on arithmetic schemes file


For an extension to curves of higher genus see

T. Oliver, Zeta functions of arithmetic surfaces and two-dimensional adelic integrals, arXiv:1311.6964  


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, D. Bump, S. Friedberg, D. Goldfeld,  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  


J Adeles on arithmetic surfaces 

- Adelic geometry 

- Analytic adeles on surfaces 

- Higher adelic zeta integral and unramified two-dimensional Iwasawa-Tate theory

- Translation invariant measure and integration on higher analytic adelic structures 


[45] 2d zeta integral, geometry of 2d adeles and the BSD conjecture for elliptic curves over global fields (Analysis on arithmetic schemes. III), work in progress pdf 

[44] Geometric adeles and the Riemann–Roch theorem for 1-cycles on surfaces, preprint 2013, to be published in 2015 pdf 

[43] Analysis on arithmetic schemes. II, J. K-theory 5 (2010), 437-557 pdf  

[42] Adelic approach to the zeta function of arithmetic schemes in dimension two, Moscow Math. J. 8 (2008), 273-317 pdf  


For some of related arithmetic and geometric aspects see 

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 local integration, harmonic analysis and zeta integrals 

- Higher Haar measure and integration, harmonic analysis on higher local fields 

- Higher local zeta integrals  

- Integration on algebraic groups over higher local fields and their representation theory

- Links with model theory and Feynman functional integration


[41] 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 pdf  

[40] Analysis on arithmetic schemes. I, Docum. Math., (2003), 261-284 pdf  


For related measure and integration aspects see  

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


For related representation theoretical aspects see  

  A. Braverman and D. Kazhdan, Some examples of Hecke algebras for 2-dimensional local fields, Nagoya Math. J. 184 (2006), 57–84 

D. Kazhdan, Fourier transform over local fields, Milan J. Math. 74 (2006), 213–225 

D. Gaitsgory and D. Kazhdan, Representations of algebraic groups over a 2-dimensional local field, Geom., Funct. Anal. 14 (2004), no. 3, 535–574 

K.-H. Lee, Iwahori-Hecke algebras of SL_2 over 2-dimensional local fields, Canad. J. Math. 62 (2010), 1310-1324   

H. Kim and K.-H. Lee, Hecke algebras of SL_2 over 2-dimensional local fields,  Amer. J. Math. 126 (2004), 1381–1399 


For related model theoretical aspects  see  

E. Hrushovski and D. Kazhdan, The value ring of geometric motivic integration and the Iwahori-Hecke algebra of SL_2, 2  Geom. funct. anal. 17 (2008), 1924–1967 

E. Hrushovski and D. Kazhdan, Integration in valued fields,  Drinfeld Festschrift, Algebraic geometry and number theory, 261–405, Progr. Math., 253, Birkhauser Boston, 2006

H Interactions of model theory, arithmetic and algebraic geometry and noncommutative geometry


[39] Model theory guidance in number theory? -  In Model Theory with Applications to Algebra and Analysis, LMS Lecture Note Series, 349, CUP, 2008, 327-334 pdf  

[38] Several nonstandard remarks, - In AMS/IP Advances in the Mathematical Sciences,  AMS Transl. Series 2, vol. 217 (2006), 37-50 pdf  

[37] Remark 1 sect. 4 and Remark 3 sect. 13 of Analysis on arithmetic schemes. I, Docum. Math., (2003), 261-284 pdf  


See also  

C. Birkar, Elements of nonstandard algebraic geometry math.AG/0303206  

L. Taylor, A nonstandard approach to real multiplication, math/0612184

L. Taylor, Higher derivatives of L-series associated to real quadratic fields, math/0612186

L. Taylor, Line bundles over quantum tori, math.NT/0612189  

B. Clare, Nonstandard mathematics and new zeta and L-functions arXiv:0808.1965 


G Arithmetic noncommutative class field theory and local reciprocity maps


[36] On the image of noncommutative reciprocity map, Homology, Homotopy and Applications, 7(2005), 53-62 pdf  

[35] 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 pdf  

[34] 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  

papers of K.I. Ikeda and E. Serbest: arXiv:0805.3431, arXiv:0805.3420, Acta Arithm.144(2010)


F Infinite ramification theory and pro-p-group theory


[33] 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  

[32] On just infinite pro-p-groups and arithmetically profinite extensions of local fields, J. Reine Angew. Mathematik 517(1999), 61-80 pdf  


See also  

papers of C. Griffin, including math.GR/0209193, math.GR/0310038


E Local field arithmetic, representation theory or algebraic groups


[31] Last section of Adelic approach to the zeta function of arithmetic schemes in dimension two, Moscow Math J 8 (2008), 273-317 pdf  

[30] I. Efrat and I. Fesenko, Fields Galois-equivalent to a local field of positive characteristic, Math. Res. Lett. 6 (1999), 345-356 pdf  


See also  

generalizing the Deligne-Lusztig theory to algebraic groups over truncated Witt victors, papers of A. Stasinski  



D Ramification theory, finite and infinite Galois extensions, perfect and imperfect residue field


[29] Ch. 3 of I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions,  Second extended edition , AMS 2002, 341 pp. 

[28] On deeply ramified extensions, Journal of the LMS (2) 57(1998), 325-335. 

[27] Hasse-Arf property and abelian extensions, Math. Nachr. 174(1995), 81-87. 

[26] Abelian local p-class field theory, Math. Ann. 301 (1995), 561-586. 


C Higher local fields, their structures, their algebraic K-groups


[25] Ch. 9 of I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions,  Second extended edition , AMS 2002, 341 pp. 

[24] Sequential topologies and quotient 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 pdf  

[23] 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

[22] 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. Cámara, Topology on rational points over higher local fields, arXiv:1106.0191

A. Cámara, Functional analysis on two-dimensional local fields, Kodai Math. J. 36 (2013),536-578 arXiv:1210.2995

A. Cámara, Locally convex structures on higher local fields, arXiv:1210.8068


B Class field theories,  one-dimensional and higher dimensional


[21] Chapter 3 of Analysis on arithmetic schemes. II, J. K-theory 5 (2010), 437-557 pdf  

[20] Ch. 4, 5 of I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions,  Second extended edition , AMS 2002, 341 pp. 

[19] 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.

[18] 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.

[17] 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. 

[16] On general local reciprocity maps, J. reine angew. Math. 473(1996), 207-222.

[15] Here is a review of various local class field theories from Abelian extensions of complete discrete valuation fields, Number Theory Paris 1993/94, Cambridge Univ. Press 1996, 47-74.

[14] Abelian local p-class field theory, Math. Ann. 301 (1995), 561-586. 

[13] Local class field theory: perfect residue field case, Izvestija Russ. Acad. Nauk. Ser. Mat. 57 issue 4(1993), 72-91; English transl. in Russ. Acad. Scienc. Izvest. Math. 43(1994), 65-81.

[12] On norm subgroups of complete discrete valuation fields, Vestn. St. Petersburg Univ. Series I, issue 2 1993, 54-57. 

[11] 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. 

[10] 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.

[9] 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. 

[8] 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


[7] Ch. 7,8 of I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions,  Second extended edition , AMS 2002, 341 pp. 

[6] The generalized Hilbert symbol in multidimensional local fields, in Rings and Modules, vyp. 2, 1988, 88-92. 

[5] 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. 

[4] Explicit formulas for the generalized Hilbert symbol on Lubin-Tate formal groups - see  Ch. I of 1987 thesis 

[3] 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. 

[2] The Hilbert symbol on Lubin-Tate formal groups. III, in Rings and matrix groups, 1984 , 146-150. 

[1] 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


[T3] Invitation to higher local fields, I. Fesenko, M. Kurihara (eds.) Geometry and Topology Monographs vol 3, Warwick 2000 

[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


[S5] Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Europ. J. Math. pdf

[S4] Adelic approach to the zeta function of arithmetic schemes in dimension two, Moscow Math. J. 8 (2008), 273-317 pdf

[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


[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