Selected open problems on higher local fields,  adeles,  zeta integrals.  

(D stands for ‘done’, P stands for ‘in progress’)

Local aspects

1. Develop a good theory of ramified local zeta integral. Its values will be in C((X)) and the meaning of non-constant terms should be clarified. Use it to construct a general higher local ramification theory. Detect its connections with existing partial approaches to higher ramification theory. 

D2. Develop invariant measure and integration on algebraic groups over higher local fields. GL(n) case done by Matthew Morrow, a new approach more directly generalisation the theory on the additive group is done by Matthew Waller. 

P2’. Develop applications to representation theory of algebraic groups over higher local fields and of Kac-Moody groups - this is in progress by Matthew Waller.

3. Understand better analogies between the Fourier transform on two-dimensional local fields and the Feynman path integral and use them in both directions.

4. Find a general conceptual theory which unifies several quite 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) nonarchimedean functional analysis. Some work in direction (f) has been done by Alberto Camara.

Adelic aspects

1. Similar to Local  Problem 4, find a unifying theory, which combines features of topological, categorical approaches and model-theoretical approaches, to deal with the adelic structures (two different: geometric and analytic adelic structures) in dimension two. 

Search for a universal geometric-analytic adelic structure which takes into account the integral structure of rank 1 and of rank 2 and is related to the study of 0-cycles and 1-cycles.

2. 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 [10]. For positive characteristic see 10. 

D3. Prove all expected topological properties of geometric adeles on algebraic surfaces using higher adelic self-duality (done in )

D4. Find a two dimensional adelic proof of the  R-R theorem for surfaces (done by IF, in 

P4’. The same text asks more open questions about the adelic algebraic geometry on surfaces, e.g. an adelic proof of the Noether formula and Hodge theory, as well as a higher-dimensional version. Students in Nottingham are working on some of these problems.

P5. Develop a more refined measure and integration which takes into account range of  coefficients of finitely many powers of the main local parameter. Wester von Urk is working on this.

6. 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 dimension two. See also 27.

P7. Following the one-dimensional general linear adelic group theory (e.g. Goldfeld-Hundley), develop appropriate elements of two-dimensional theory. For the local theory, try to develop two dim analogues of 2d local Whittaker functions, 2d Kirillov model, 2d Jacquet model. Matthew Waller is working on this. See also 27.

8. Find  a more general theory of  measure and integration on T for the purposes of the zeta integral. 

D9. Develop a theory of (renormalized) measure and integration on analytic adelic spaces for models of curves of genus >1. Done by Tom Oliver. 

D10. Develop a positive characteristic class field theory theory extending the method of Kawada-Satake. Done by Kirsty Syder. See also 2. 

11. Develop elements of an enhanced two-dim 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  specializes both to geometric adeles and to analytic adeles. 

12. Two-dim theta formula - understand it better from several directions, and try to get an enhanced algebraic geometric proof of two-dim theta formula and which can be used to deal with the zeta integral in a more categorical or geometric way. See also 11. Try to investigate if there are other summation formulas/theta formulas which can be used in the study of the zeta integral.

13. Compute the zeta integral for an arbitrary function f in two-dim B-S space.

14. Develop the theory of zeta integrals for singular points on fibres of general type and establish comparison with the zeta function.

15. Apply the previous for an adelic interpretation of the conductor, see Remark 2 in sect. 40 of Analysis II. Obtain an adelic understanding of the wild part of the conductor. 

16. Develop a theory of ramified (\chi_0\not=1) 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 Local-1. 

D17. Develop the theory of zeta integral on models of curves of higher genus, following the general scheme outlined in section 57, see also 9. Done by Tom Oliver. 

18. 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 and Suzuki-Ricotta-F paper.

19. Further develop the new correspondence zeta functions <-> mean-periodic functions and various related things, including connections with the Langlands correspondence. See also 34.

20. Find more applications of mean-periodicity, in particular using Suzuki-Ricotta-F paper and other papers of Masatoshi Suzuki. 

21. Using adelic geometry progress towards hypothesis (*) in section 51, closely related to the GRH.

22. Develop the theory sketched in section 55 of [10]. 

23. Investigate the direction of Remark 1 in section 56 of [10], 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. 

24. 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).

D25. Analogously to the positive characteristic case compare the arithmetic rank of an elliptic curve over a global fields and the Picard rank of the model in characteristic zero case and write down the details. Done by Matteo Tamiozzo

26. Find an adelic approach to Arakelov intersection pairing on model of elliptic curves and its applications including a proof of the discreteness of the function field. 

27. Following the outline in the last section of Adelic approach to zeta functions develop a 2 dim adelic theory of automorphic functions and representations. See also 30. 

28. Using the objects which naturally come from the theory of two dimensional zeta integral understand and develop a "correct" theory of bundles on arithmetic surfaces extending the one-dimensional classical observation of Weil. See also 31. 

29. 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 noncommutative theory for L-factors of the zeta function. Analytically we already have many relations, the issue is to get them algebraically and geometrically. 

30. Develop the theory of Eisenstein series on arithmetic surfaces.

31. 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 19 and 27. 

32. Find possible relations between the two dimensional theta formula and other recent "non-commutative" summation formula by Laurent Laffogue.

33. In positive characteristic find a purely adelic proof of the full BSD conjecture without using the previous results proved by other techniques (Tate, Artin, Milne). 

34. 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.

35. Find a two-dimensional adelic description of an arithmetic analogue of the Bogomolov-Miyaoke-Yau inequality and its applications. 

36. Further developing Problem 15, find an adelic interpretation of the discriminant inequality. 

37. Find more explicit relations between the theory of two adelic structures on arithmetic surfaces and their applications to the study of the zeta functions and the two symmetries of IUT and Hodge theatres in IUT, to possibly enhance each of the theories.