Based on the analogy between number and function rings, for a fixed prime $p$, delta geometry is developed based on the notion of a $p$-derivation $\delta$ which plays the role of 'differentiation' for number rings. Such a $p$-derivation $\delta$ comes from the $p$-typical Witt vectors and as a result, delta geometry naturally encodes valuable arithmetic information, especially those that pertain to lifts of Frobenius.

For an abelian scheme, using the theory of delta geometry, one canonically attaches a filtered isocrystal that bears a natural map to the crystalline cohomology. In this talk, we will examine the construction and properties of the above filtered isocrystal and also discuss some comparison results that explain its relation to crystalline cohomology.

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