# Motivation behind Fontaine's Theory

I am reading Fontaine's theory of $$p$$-adic Galois representations. But I am not able find the motivation behind it. Please let me know some good reference where I can study the motivation behind Fontaine Theory.

• Not sure what exactly you mean with "Fontaine's theory", but for instance the motivation for the period rings is that we would like to have comparison theorems like those over $\mathbb C$. For various reasons $\mathbb C_p$ is not good enough for this purpose (since there is no $p$-adic $2\pi i$). – Wojowu Nov 27 '19 at 13:32

Fontaine's program is the classification of $$p$$-adic representations of $$\operatorname{Gal}(\bar{K}/K)$$ where $$K$$ is a discrete valuation field of residual characteristic $$p$$.

If by motivation, you meant why Fontaine himself wanted to do that, you could do worse than reading

https://webusers.imj-prg.fr/~pierre.colmez/FW.pdf

and/or

https://www.math.u-psud.fr/~illusie/Illusie-Pisa5.pdf

depending on whether your background is in arithmetic or algebraic geometry.

If by motivation, you meant why we (for some value of we) might want to do this, well you could have a look at the Proceedings of the ICM of 2002* for instance, look up the lectures which mention Fontaine's theory and what they deal with (I found 5 different ones with topics ranging from special values of $$L$$-function to $$K$$-theory to Galois representations and automorphic forms).

*I chose the year rather arbitrarily, I'm guessing that any other years since 1984, and certainly 2018, would do.