(D. Isaksen) Compute the motivic Steenrod algebra for more general base schemes. (Hard; see Hoyois-Kelly-Østvær for the latest)
(D. Isaksen) Construct a “motivic modular forms” spectrum. (Hard; Østvær may have some ideas about this)
(D. Isaksen) Study $p$-complete motivic homotopy theory over fields of characteristic $p$. (Hard)
(D. Isaksen) Find an algebraic model for rational motivic homotopy theory.
(D. Isaksen) Uniqueness of motivic homotopy theory (following Schwede).
(D. Isaksen) Compute motivic stable homotopy groups at odd primes. (This might not be very interesting)
(A. Salch) The Ravenel conjectures in nonclassical settings–equivariant, motivic, and/or parametrized spectra.
Chromatic convergence can be reduced to some kind of understandable criterion on the homotopy-theoretic setting you’re in (you need a homotopy-commutative monoid object similar enough to $\mathrm{BP}$) and represents work in progress. But the others are pretty wide open. The localization conjecture (that ${\mathrm{BP}}_{*}({L}_{E(n)}X)={v}_{n}^{-1}{\mathrm{BP}}_{*}(X)$ if $X$ is $E(n-1)$-acyclic) seems to be one of the more doable ones and it would strengthen chromatic convergence.
The others are probably hard, especially nilpotence, because the obvious analogue just isn’t true: in motivic spectra over $\mathrm{Spec}\u2102$, for example, there’s an infinite (nonvertical) tower in the Adams-Novikov ${E}_{\mathrm{\infty}}$ term for the sphere. It’s generated by one of the two analogues of $\eta $ in the motivic setting. Dugger and Isaksen conjecture that all the infinite towers in the ANSS ${E}_{\mathrm{\infty}}$-term for the sphere over $\mathrm{Spec}\u2102$ are generated by those $\eta $-multiplications on various elements–or, in other words, if you choose a homotopy-commutative ring spectrum such that $\mathrm{BPGL}$ is an $E$-algebra, and that $\eta $ is in the Hurewicz image in ${\pi}_{*}(E)$ (so that $\eta $ is on the 0-line in the $E$-Adams SS), then nilpotence should hold in the $E$-Adams SS for motivic spectra over $\mathrm{Spec}\u2102$. The trick is finding such a ring spectrum $E$ which also has all the nice properties (e.g. the localization conjecture) that $\mathrm{BP}$ has–otherwise you could just let $E=\mathbb{S}$. This makes it seem like nilpotence is interesting and worth looking into in these contexts.
A general approach to Ravenel conjectures in an arbitrary “nice” (compactly generated) symmetric tensor triangulated category $C$ is to look at the poset of thick subcategories $T$ of the full subcategory generated by the compact objects, with the following special property: you want only the thick subcategories $T$ such that the localization killing $T$ is smashing on the entire triangulated category $C$. That’s actually a poset (there’s no set-theoretic trouble) due to generalizations of Ohkawa’s theorem–Dwyer and Palmieri did a version of this, and Casacuberta, Gutierrez, and Rosicky did another. Anyway one way of stating the classical smashing and thick subcategory conjectures/theorems is that, when $C$ is the homotopy category of $p$-local spectra, this poset is isomorphic to the natural numbers.
From this point of view the localization conjecture + Landweber’s classification of invariant primes becomes: there exists a homotopy commutative monoid object $E$ in $C$ with the properties that:
the poset of invariant prime ideals of the Hopf algebroid $({E}_{*},{E}_{*}E)$ is isomorphic to this poset, and
for any pair $T\prime <T$ in this poset such that there is no intermediate element between $T\prime $ and $T$, any $T\prime $-acyclic object $X$ satisfies:
where ${D}_{I}$ is the ideal transform with respect to the invariant prime ideal $I$ of ${\pi}_{*}(E)$ associated to $T$. Under good conditions that ideal transform is just the localization inverting $I$.
It sounds like, in every case where we’ve been been able to compute anything, the thick subcategories are still just the Johnson-Wilson acyclics. But these are all motivic spectra over $\mathrm{Spec}$ of various fields. Motivic spectra over things with actual geometry might have more exotic smashing thick subcategories, or maybe not! Either way the answer seems interesting.
(A. Salch) Speaking of motivic spectra over actually interesting bases: it would be really good to use that stuff for what it’s actually designed for, to compute the homotopy types of the motivic spectra associated to some varieties with interesting homological/$L$-function-theoretic invariants. Examples:
The “right way” to phrase techniques and results in period theory and Dwork theory like those of Katz’s thesis is with motivic spectra. Katz looks at the period map from ${M}_{1,1}$ to ${\mathbb{P}}^{1}$ and shows that going around a loop in ${\mathbb{P}}^{1}$ produces an interesting monodromy action on the fibers of the period map and thusly an interesting operator on $\ell $-adic cohomology of elliptic curves, and he uses a Lefschetz fixed-point approach to describe the zeta-function of a curve in terms of the trace of this operator on the cohomology on the curve. In other words he has a family of motivic spectra parametrized over the complex projective line and going around a loop in the base space does something extremely interesting to the motivic spectra. How do we phrase those ideas from period theory in stable-homotopic language? I think they should translate completely.
Doug Ulmer’s work on elliptic curves, over function fields, of arbitrarily large rank. This proceeded by constructing an elliptic surface, i.e., a surface projecting down to a curve such that the fibers are each elliptic curves, and showing that arbitrarily large ranks occur among the fibers in that particular case. Again, since rank is supposed to be visible from the motive, it seems like something interesting is happening here if you regard this as a family of motivic spectra parametrized over that same base curve, or as a single motivic spectrum (of the whole surface) over the base field. How do the Adams spectral sequences for computing the various homological invariants of the curves from their motivic spectra vary as you move around from fiber to fiber on that base curve, etc.?
Mike Hopkins and his collaborators have started a program to study algebraic vector bundles. This program led them to make the following hypothesis which would imply many results about algebraic vector bundles. “Wilson Space Hypothesis”: The motivic space ${\Omega}^{\mathrm{\infty}}{S}^{2n,n}\wedge \mathrm{MGL}$ has a cell decomposition with only cells of the form ${S}^{2k,k}$. (Here ${S}^{2k,k}={\mathbb{P}}^{k}/{\mathbb{P}}^{k-1}$.) So far the ${C}_{2}$-equivariant analogue of this statement for $\mathrm{MU}\mathbb{R}$ has been proven by Hill-Hopkins, which one can take as evidence.