et  be a commutative Noetherian ring,  and  two ideals of  and  a finite -module. In this paper, we have studied the vanishing and relative Cohen-Macaulyness of the functor for relative Cohen-Macauly filtered modules with respect to the ideal  (RCMF). We have shown that the for relative Cohen-Macaulay modules holds for any relative Cohen-Macauly module with respect to  with ........

A “diophantine equation” is a system of polynomial equations where we restrict the solution set to rational or integral values. The study of diophantine equations is the study of the nature of these integral or rational solution sets. “diophantine geometry” is the study of diophantine equations using the language of algebraic geometry. Specifically, one associates to a diophantine equation the ...

In this paper, we consider an arithmetic Hodge index theorem for a family of semi-stable curves, generalizing Faltings-Hriljac’s arithmetic Hodge index theorem for an arithmetic surface.

‎The concept of self-similarity on subsets of algebraic varieties‎ ‎is defined by considering algebraic endomorphisms of the variety‎ ‎as `similarity' maps‎. ‎Self-similar fractals are subsets of algebraic varieties‎ ‎which can be written as a finite and disjoint union of‎ ‎`similar' copies‎. ‎Fractals provide a framework in which‎, ‎one can‎ ‎unite some results and conjectures in Diophantine g...

7 Faltings’ Theorem 36 7.1 Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 7.2 Height Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.3 Removing the Nonvanishing Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.4 Proof of Nonvanishing . . . . . . . . . . . . . . . . . . . . . . . . . . . ...