Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $\lambda$-adic representations of the \'etale fundamental group of $X$ with connected algebraic monodromy groups $\bG_\lambda$, we construct a common $E$-form $\bG$ of all the groups $\bG_\lambda$ and in the absolutely irreducible case, a common $E$-form $\bG\hookrightarrow\GL_{n,E}$ of all the tautological representations $\bG_\lambda\hookrightarrow\GL_{n,E_\lambda}$. Analogous rationality results in characteristic $p$ assuming the existence of crystalline companions in $\mathrm{\textbf{F-Isoc}}^{\dagger}(X)\otimes E_{v}$ for all $v|p$ and in characteristic zero assuming ordinariness are also obtained. Applications include a construction of $\bG$-compatible system from some $\GL_n$-compatible system and some results predicted by the Mumford-Tate conjecture. (If you would like to join this seminar please contact Bo-Hae Im to get the zoom link.)

A theorem of Khare-Wintenberger and Kisin (once Serre’s modularity conjecture) says that every two-dimensional odd absolutely irreducible representation \bar\rho of the Galois group of the rationals over a finite field comes from a modular form f, that is, \bar\rho ~ \bar\rho_f. The conjecture even provides a recipe for the weight, level and character of f, but does not give any information about the slope of f. In this talk, based on joint work with Kumar, we provide conditions on f - the main one being that the weight of f is close to 0 - which guarantee that the slope of a modular form g giving rise to the twist of \bar\rho_f by the cyclotomic character has slope one more than the slope of f. This provides a global explanation of some local patterns mentioned in our first talk. The proof uses the theta operator and Coleman-Hida families of overconvergent forms. (This is the second of the two KAIX Invited Lectures.)

The zig-zag conjecture predicts that the reductions of two-dimensional irreducible p-adic crystalline representations of half-integral slope and exceptional weights - weights which are two more than twice the slope modulo (p-1) - have reductions which are given by an alternating sequence of irreducible and reducible representations. Some partial progress was made towards this conjecture over the years by Buzzard-Gee (slope 1/2), Bhattacharya-G-Rozensztajn (slope 1) and G-Rai (slope 3/2). In this talk I shall use work of Breuil-Mézard and Guerberoff-Park in the semi-stable case and a limiting argument connecting crystalline and semi-stable representations due to Chitrao-G-Yasuda to show that zig-zag holds for half-integal slopes bounded by (p-1)/2, at least on the inertia subgroup, if the weight is sufficiently close to a weight bounded by p+1. (This is the first of the two KAIX Invited Lectures.)