Birth of "crystals" - Grothendieck's quote: “I’m reading Manin’s ... I think one should introduce the notion of slope, and Newton polygon”

Question

The must-read article on Grothendieck's epoch: "Reminiscences of Grothendieck and His School" by Luc Illusie ("chatted by the fireside, recalling memories of his days with Grothendieck" to Alexander Beilinson, Spencer Bloch, Vladimir Drinfeld, et al.)

There is a quote there:

"I remember, once he (Grothendieck) said, “I’m reading Manin’s paper on formal groups and I think I understand what he’s doing. I think one should introduce the notion of slope, and Newton polygon,” then he explained to us the idea that the Newton polygon should rise under specialization, and for the first time he envisioned the notion of crystal".

Question: I wonder if someone might comment on what is going on here: Newton polygon of what? What specialization ? What slope ? And why it is related to crystals ?

It seems that nowadays some considerations similar to that are familiar to experts (moreover their q-analogs also considered). E.g. here: "MODULAR q-HOLONOMIC MODULES" (S. GAROUFALIDIS, C. WHEELER) (pages 24,31,32-38).

Let me add more quote:

"Then at the same time, maybe, or a little later, he wrote his famous letter to Tate: “… Un cristal possède deux propriétés caractéristiques : la rigidité, et la faculté de croître, dans un voisinage approprié. Il y a des cristaux de toute espèce de substance: des cristaux de soude, de soufre, de modules, d’anneaux, de schémas relatifs, etc.” (“A crystal possesses two characteristic properties: rigidity, and the ability to grow in an appropriate neighborhood. There are crystals of all kinds of substances: sodium, sulfur, modules, rings, relative schemes, etc.”)"

Answer 1

Presumably this is Manin's paper "The theory of commutative formal groups over fields of finite characteristic". In this paper, formal groups are classified using Dieudonné modules, and torsion-free Dieudonné modules over a field are a special case of F-isocrystals over the field, whose importance Dave Benson already highlighted. (This is because Dieudonné modules are expressed in terms of operators $F$ and $V$ while $F$-isocrystals only have $F$. In the torsion-free case, $V$ is determined by $F$, but $V$ may not exist for all possible $F$).

Grothendieck was very attuned to the idea that one can figure out how to define cohomology groups in arbitrary degree by carefully studying $H^1$. In the case of étale cohomology, one of his key insights (simplifying greatly) was that if the étale fundamental group can be used to give a purely algebraic definition of $H^1$ of a variety mimicking the topological $H^1$ then the same notion could be used to define all the cohomology groups.

Grothendieck presumably saw that (the dual of) the formal group of an abelian variety behaves like $H^1$ of that abelian variety for some as-yet-undefined cohomology theory, and therefore can be used to define $H^1$ of an arbitrary variety (using the Albanese), and therefore wondered what the higher $H^i$ of the cohomology theory would look like. This required a suitable generalization of Dieudonné modules with generalizations of the phenomena that occur already for them, e.g. their slopes and Newton polygons.

The key specialization statement is that for a family of abelian varieties, there is a stratification of the base of the family with the Newton polygon of the formal group constant on each stratum, and the Newton polygon on a stratum lies below the Newton polygon on any stratum in the closure of that one. To generalize this statement one needs a category of objects over an arbitary base scheme which admit such a Newton polygon of Frobenius at every point, which eventually was realized in the category of $F$-isocrystals.