4 problems

Manin's conjecture. This $\Gamma$-factor should be the zeta function of the dual of $\mathbb{P}^{\infty}_{\mathbb{F}_1}$.

Let $X$ be a smooth scheme, and let $K(X)$ denote its $K$-theory spectrum, regarded as a module over the $K(S)$ ring spectrum. Compactness or dualizability conjecture. Then $K(X)$…

Let $k$ be a field, and let $K(\mathbf{Var}_{/k})$ be the $K$-theory spectrum of varieties over $k$. Let $K(S)$ be the $E_\infty$-ring spectrum whose modules contain the $K$-theory…

Let $k$ be a field. Write $K(\mathbf{Var}_{/k})$ for the $K$-theory spectrum of varieties over $k$, and let $K(S)$ and $K(k)$ denote the relevant $K$-theory $E_\infty$-ring spectra…