The space of all Dirichlet series convergent in some half-plane should be a countable-dimensional C-VS. Is there a Hilbert space structure? Perhaps we should consider some other, but similar space instead, such as the log of such fns. In any case, there are lots of things you could do with such a space. Add and multiply functions. Convolution? Pointwise (coordinatewise) multiplication? Differentiation? Shifting? If Hilbert space structure, what operators can we consider, and how will the spectrum of these look like? What happens to functions in these space when we do things on corresponding motivic or automorphic objects? Or on dynamical systems? What invariants can be attached to the functions, and how do they behave under various operations? Special values? Limits for analytic continuation? Order of vanishing? If a motivic L-functions (perhaps with some additional data) determines the motive, can we define a motive (or perhaps even a mixed motive) to BE an L-function (possibly with some additional data)??? What interesting subspaces (in some generalised sense?) of the above space should we consider. Those with coeffs in some number field? Those coming from a certain class of objects? Those with Euler prods? Those satisfying RH? Those satisfying some special value conjecture? What are the dimensions (if well-defined) of these sets? There will be some functions coming from function fields. How does this set relate to other sets, e.g. functions coming from number fields? Is there a notion of “degree” of a function in our space - something that would agree with the dim of the adelic group or the dimension of the Galois rep? If there is, it would probably be related to Euler prods, but not well-defined for all functions in the space. Call our space H. What about corresponding p-adic spaces? Would the inverse Mellin transforms of elts of H live in some space which can be completed p-adicly to get p-adic modular forms etc? Could we complete directly our space H to get p-adic L-functions (this is too naive, but perhaps modify???). Is there a nice transform mapping the critical strip to a bounded set, so that it becomes easier to numberically verify RH?
There exists “analytic function theory for operatorvalued functions of one complex variable”. (Davies, p. 4) Can we consider L-functions which are operator-valued? Can such an operatorvalued L-function somehow give rise to a family of ordinary L-functions?
interval methods in p-adic analysis?
Transform methods applied to L-functions? ‘L-functions’ of several variables? Can these satisfy partial diff equations? Do ordinary L-functions satisfy any nice diff equations?
For GRH, can consider real and imag parts of zeta separately. these are harmonic, both must vanish simultaneously. what about zeros of eg real part in complex plane?
note: ‘harmonic implies analytic’ (evans 2.2)
There are various notions of modular form more general than the one discussed above. The assumption of complex analyticity can be dropped; Maass forms are real-analytic eigenfunctions of the Laplacian but are not holomorphic. Groups which are not subgroups of SL2(Z) can be considered. Hilbert modular forms are functions in n variables, each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in a totally real number field. Siegel modular forms are associated to larger symplectic groups in the same way in which the forms we have discussed are associated to SL2(R); in other words, they are related to abelian varieties in the same sense that our forms (which are sometimes called elliptic modular forms to emphasize the point) are related to elliptic curves. Automorphic forms extend the notion of modular forms to general Lie groups.
Ideas: motivic L-funcs
What happens with the L-function when the motive varies “continuously” over some moduli space, in particular, taking the moduli space to be a modular curve so that we easily can compute invariants of the L-functions?
Can naively view a motive as a var given by a system of dioph eq’s. coeffs of these eqs -> combinatorics. combinatorial invariants vs cohom’l invar’s, invar’s of the L-function?
is it true that ‘every L-func from geom is def by euler product, ie multiplicatively? eg can def L-f in terms of sol counts of ECs, ie additively. generalise this?
Can we try to construct a motive for which RH holds?
What happens with the L-function when the motive varies “continuously” over some moduli space? What happens with the zeroes? In particular, if the zetafunction has a pair of zeroes off the critical line, can we then construct a family of motives “close to Spec Z” with the same property?
Can we use Mori program to get a better understanding of cats of motives? Is Mori only for C or also other fields??
We want to think about the cat of all cohomology theories, in some suitable sense (and also homology theories). This cat should somehow correspond to a cat of “sheaf costructions”, or sites, in which for example ordinary cohom corresponds to the Zariski site. However, we must generalize the notion of a site since for example l-adic cohom does not really correspond to a site in the strict sense. The universal property of motivic cohom should translate into a universal property for “motivic sheaves”. We should also be able to transfer operations between cat of cohom theories and cat of “sites”.
nLab page on Old ideas