Lattices in Tate modules.

Area of Science:

• Algebraic Geometry
• Number Theory

Background:

• Zarhin's theorem provides a framework for understanding endomorphisms of abelian varieties.
• Abelian varieties and their associated modules are fundamental objects in arithmetic geometry.

Purpose of the Study:

• To refine Zarhin's theorem concerning the action of endomorphisms on modules of abelian varieties.
• To establish a matrix representation for endomorphisms on Tate and Dieudonné modules.

Main Methods:

• The study involves refining existing theorems in algebraic geometry.
• It utilizes the structure of Tate modules and covariant Dieudonné modules.

Main Results:

• A matrix representation for the action of an endomorphism 'u' on the Tate module of a 'g'-dimensional abelian variety 'X' is proven to exist.
• A similar matrix representation is shown for the covariant Dieudonné module over a perfect field of characteristic 'p'.

Conclusions:

• The findings generalize and strengthen previous results regarding endomorphisms of abelian varieties.
• This work provides a more concrete understanding of the algebraic structure of these objects.