Ivan Fesenko
1. Develop a good theory of 2d ramified local zeta integral. Its values will be in R((X)) and the meaning of non-constant terms should be clarified. Use it to construct a new 2d ramification theory. Find its connections with the existing partial approaches to higher ramification theory.
2. Using the theory of translation invariant measure and integration on the additive and multiplicative groups of 2d local fields, develop a theory of translation invariant measure and integration on algebraic groups over 2d local fields. GL(n) case was done by M. Morrow. A new approach more directly generalising the theory on the additive and multiplicative group in Analysis I is done by R. Waller.
3. Develop applications to representation theory of algebraic groups over 2d local fields - some work in this direction is done by R. Waller.
4. Understand better analogies between the Fourier transform on 2d local fields and the Feynman path integral and use these analogies in both directions.
5. Find a general theory that unifies different existing approaches to higher local fields and their arithmetic such as (a) topological and sequential topological, (b) higher translation measure theoretical, (c) iterated ind-pro categories, (d) higher categorical, (e) model theoretical, (f) non-archimedean functional analysis. Some work in direction (f) was done by Alberto Camara.
6. Find a unifying theory that combines features of topological, categorical approaches and model-theoretical approaches, to deal with the two different adelic structures on surfaces: geometric and analytic adelic structures. Find a universal geometric-analytic adelic structure which takes into account the integral structure of rank 1 and of rank 2 and in particular is related to the study of 0-cycles and 1-cycles.
7. Develop explicit global and semi-local-global class field theories for arithmetic surfaces, using the explicit higher local class field theory a la Neukirch, along the description in Analysis II.
8. Develop a 2d adelic approach to Arakelov geometry and the Deligne pairing. Partially done by P. Dolce and W. Czerniawska.
9. Develop a more refined measure and integration which takes into account the range of coefficients of finitely many powers of the main local parameter. Partially done by R. Waller.
10. Using the local and adelic theories for GL_1, and the measure and integration for local GL_n, develop measure and integration on GL_n(A), A the analytic adeles and its application to automorphic representations in 2d. See also 26.
11. Generalising the 1d general linear adelic group theory (e.g. Goldfeld-Hundley), develop appropriate elements of 2 theory. For the 2d local theory, develop 2d analogues of local Whittaker functions, Kirillov model, Jacquet model. See also 26.
12. Develop elements of an enhanced 2d algebraic geometry which takes into account zero cycles and integral structures of rank 2 on surfaces and which could possibly help to find a more universal adelic object which specialises both to geometric adeles and to analytic adeles.
13. Understand 2d theta formula better, from several directions. Try to get an enhanced algebraic geometric proof of 2d theta formula, which can be used to deal with the zeta integral in a more categorical or geometric way. Try to investigate other summation formulas/theta formulas which can be used in the study of the zeta integral.
14. Develop the theory of zeta integrals for singular points on fibres of general type and establish its comparison with the zeta function.
15. Produce a 2d adelic interpretation of the conductor, see also Remark 2 in sect. 40 of Analysis II. Obtain an adelic understanding of the wild part of the conductor.
16. Develop a theory of 2d ramified adelic zeta integral, clarify the meaning of non-free coefficients of ramified zeta integrals and use them to obtain more information about ramification invariants. See also 1.
17. In the context of the correspondence: zeta functions <-> mean-periodic functions, study the mean-periodicity of the boundary function H in the space of smooth functions of exponential growth on the real line, see section 48 of Analysis II and Suzuki-Ricotta-F paper.
18. Develop further the new correspondence zeta functions <-> mean-periodic functions, including connections with the Langlands correspondence. See also 33.
19. Find new applications of mean-periodicity, in particular using Suzuki-Ricotta-F paper and other papers of Masatoshi Suzuki.
20. Progress towards a proof of hypothesis (*) that is closely related to the GRH, in section 51 of Analysis II.
21. Develop the theory sketched in section 55 of Analysis II.
22. Investigate the direction of Remark 1 in section 56 of Analysis II, a 2d generalisation of the Weil-Connes approach to the study of the zeta function and zeta integral and its applications to their meromorphic continuation and functional equation.
23. As part of the study of 2d class field theory, develop further the K_1 times the Brauer group theory for arithmetic surfaces of Shuji Saito and extend it to the general case (without the restriction of absence of real places).
24. Find possible applications of 2d zeta integrals to Langlands correspondences. (this is related to problems of Class field theory, its three main generalisations, and applications).
25. Find possible relations between the two dimensional theta formula and other recent ‘non-additive’ summation formula on adelic algebraic groups by L. Laffogue.
26. Following the outline in the last section of Adelic approach to zeta functions develop a 2d adelic theory of automorphic functions and representations. See also 30.
27. Using the objects which naturally come from the theory of two dimensional zeta integral understand and develop an enhanced theory of bundles on arithmetic surfaces extending the one-dimensional classical observation of Weil.
28. Various problems on relations between the two dimensional commutative theory of the zeta functions of models of elliptic curves over global fields and one dimensional non-commutative theory for L-factors of the zeta function. Analytically we already have many relations, the issue is to get them algebraically and geometrically.
29. Develop the theory of Eisenstein series on arithmetic surfaces.
30. Various problems on relations between the two dimensional commutative theory of the zeta functions of arithmetic surfaces in positive characteristic and aspects of geometric Langlands correspondence. One analogy between the two theories is that each reduces the analytic aspects of the zeta/L functions to adelic geometric or geometric aspects. See also 18 and 26.
31. In positive characteristic, using Analysis III, find a purely adelic proof of the full BSD conjecture, i.e. without using the previous results proved by other techniques of Tate, Artin and Milne.
32. Following Analysis III, progress towards the relation of the analytic and arithmetic/geometric ranks of a regular model of an elliptic curve over a global field at the central point. Partially done in Analysis III.
33. Develop a general ramification theory for surfaces compatible with 2d CFT and 2d zeta integral, and taking into careful account ramification theory at the one-dimensional residue level. (this is Problem 5 of Class field theory, its three main generalisations, and applications).
34. Develop a special 2d CFT which uses torsion structures, to provide new insights into 2d CFT. (this is Problem 6 of Class field theory, its three main generalisations, and applications).
35. Find new relations between 2d CFT, anabelian geometry and Langlands correspondences. (this is part of Problem 7 of Class field theory, its three main generalisations, and applications).