p-adic cohomology and Langlands' correspondence

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In this work, I tried to construct a foundational theory of p-adic cohomology. First, I proved the Cebotarev density type theorem, which states that we can retrieve p-adic differential equation when we look at the Frobenius action, and established the theory of weights in p-adic cohomology theorem as well as the purity for the intersection cohomology. These results are expected to be applied to the “petits camarades conjecture” by Deligne claiming that the p-adic and l-adic cohomology theories have essentially the same information.