Research Texts   


 A variety of  seminars, lectures and video lectures related to the following texts



K Studying zeta functions of arithmetic surfaces via  higher adelic zeta integrals
- Meromorphic continuation of zeta functions and mean-periodicity correspondence
- Positivity hypothesis and the GRH  for zeta function of regular models of elliptic curves
- Interaction between two adelic structures on surfaces and  the zeta function at 1

J Adelic structures on arithmetic surfaces
- Geometric and analytic adelic structures on surfaces
- Translation invariant measure and integration on higher analytic adelic structures
- Higher adelic zeta integral and unramified two-dimensional Iwasawa-Tate theory


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

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

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

Geometric adeles and the Riemann–Roch theorem for 1-cycles on surfaces pdf 

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

Adelic approach to Arakelov geometry and applications, in progress pdf

 

See also  

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, Positivity of certain functions associated with analysis on elliptic surfaces, J. Number Theory 131 (2011), 1770-1796 

M. Suzuki, Two-dimensional adelic analysis and cuspidal automorphic representations of GL(2), pp.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, 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  


For some of related arithmetic and geometric aspects see 

M. Morrow, An explicit approach to residues on and dualizing sheaves of arithmetic surfaces, New York J. Math., 16(2010), 575-627 

M. Morrow, Grothendieckʼs trace map for arithmetic surfaces via residues and higher adeles, Algebra & Number Th., 2012, 6-7 (2012), 1503-1536

O. Bräunling, Two-dimensional ideles with cycle module coefficients, arXiv:1101.0424

O. Bräunling, Adele residue symbol and Tate's central extension for multiloop Lie algebras, arXiv:1206.2025



I Higher local theory (also used in J and K):
- 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


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

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  


For related aspects see  

more on measure and integration

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

representation theoretical aspects

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

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

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 

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

more model theoretical aspects

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

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 

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


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

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

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  


See also  

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

L. Taylor - A Nonstandard approach to real multiplication math/0612184, Higher derivatives of L-series associated to real quadratic fields math/0612186, 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


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  

Local reciprocity cycles, in Invitation to higher local fields, I. Fesenko, M. Kurihara (eds.) , Geometry and Topology Monographs, Warwick 2000, pp. 293-298. 

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


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


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

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  


See also  

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


E Local field arithmetic, representation theory or algebraic groups


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


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


See also  

generalizing the Deligne-Lusztig theory, papers of A. Stasinski  



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


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

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

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

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


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


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. 

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  

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

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


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,  local and higher 


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.

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

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

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.

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

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. 

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. 

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.  

On class field theory of multidimensional local fields of positive characteristic, Adv. Sov. Math. 4(1991), 103-127. 

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

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. 

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.

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.

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


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


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). 

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

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. 

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

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. 

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

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



Textbooks


I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions, A Constructive Approach,  AMS 1993, 284 pp.

I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions,  Second extended edition , AMS 2002, 341 pp.


Surveys


Abelian extensions of complete discrete valuation fields, Number Theory Paris 1993/94, Cambridge Univ. Press 1996, 47-74

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. 

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. 


Edited volumes


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


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.)

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.)

Documenta Mathematica Volume dedicated to J.H. Coates, 2006, 826 pp., I. Fesenko, S. Lichtenbaum, B. Perrin-Riou, P. Schneider (eds.) 

Documenta Mathematica Volume dedicated to A.A. Suslin, 2010, 723 pp., I. Fesenko, E. Friedlander, A. Merkuriev, U. Rehmann, (eds.) 




Several files are updated versions of the published versions