Is there a version of the Bloch–Beilinson conjecture for smooth projective varieties over global fields of positive characteristic? The conjecture I’m referring to is the “recurring fantasy” on page 1 of Bloch - Algebraic cycles and values of L-functions, though I’m particularly interested in the refined form of the conjecture, stated as conjecture 2.1.1 in Bertolini, Darmon, and Prasanna - $p$-adic $L$-functions and the coniveau filtration on Chow groups. If so, what evidence exists for the conjecture?
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This may not be precisely what you want, but a function field analogue of Beilinson's conjectures is formulated in R. Sreekantan, Non-Archimedean regulator maps and special values of $L$-functions, Cycles, motives and Shimura varieties, 469–492. Tata Inst. Fund. Res. Stud. Math., 21. Published for the Tata Institute of Fundamental Research, Mumbai, 2010.
Note as well that in W. Raskind, Higher l-adic Abel-Jacobi mappings and filtrations on Chow groups, Duke Math. J.78(1995), no.1, 33–57 it is shown that if $X$ is a smooth projective variety of dimension $n$ over a function field $k$ in one variable over a finite field of characteristic $p\neq\ell$, and such that $X$ has a proper regular model over $\mathcal{O}_{k}$, then the $\ell$-adic Abel-Jacobi map $d_{2}^{n}:\mathrm{Fil}^{2}\mathrm{CH}^{n}(X)\otimes\mathbb{Q}\rightarrow H^{2}(k,H^{2n-2}(\overline{X},\mathbb{Q}_{\ell}(i)))$ is zero.
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$\begingroup$ Thank you for the suggestion. This is certainly relevant. I will wait a bit to accept to see if there are other answers. However, I must confess that I do not know enough about higher Chow groups to determine if this conjecture implies the one we get if we just rephrase conjecture 2.1.1 in Bertolini, Darmon and Prasanna in terms of varieties over function fields. It seems difficult to check, but do you have any insight into whether this might be true? $\endgroup$– BmaCommented May 13 at 18:17
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$\begingroup$ When $X$ is a variety over a number field, the BB conjecture as stated in Bertolini et al, section 2.1 predicts the order of vanishing of $L(H^{2j-1}(\overline{X}, \mathbb{Q}_\ell), s)$ at $s = j$. In Sreekantan's notation, this function is $\Lambda(X, s)$ as defined at the start of section 7, where we take $q = 2j$. Sreekantan's conjectures seem to only predict the order of vanishing at $s = a$ when $q - 2a \ge 1$, whereas Bertolini et al applies to the case $q - 2a = 0$. So they don't seem to directly overlap. $\endgroup$– BmaCommented May 13 at 19:21
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$\begingroup$ Yes I think that’s right, Sreekantan’s article only looks at the case $q-2a\geq 1$. This is because the regulator he constructs basically comes from the boundary map in the localisation sequence. I don’t know a reference for a similar style conjecture when $q-2a=0$ I’m afraid. $\endgroup$ Commented May 13 at 20:14