Iwasawa theory, originally developed from the study of L-functions and the structure of class groups, has become a cornerstone of modern number theory. In this talk, I will focus on the Iwasawa theory of elliptic curves, delving into some of the profound conjectures that shape the field. By applying techniques from the arithmetic statistics of elliptic curves, we can investigate these conjectures "on average." The statistical study of elliptic curves reveals patterns and behaviors in large families, offering new insights that may lead to partial resolutions or alternative perspectives on long-standing open problems. If time allows, I will also explore how these investigations contribute to broader developments in arithmetic geometry and their implications for Hilbert's Tenth Problem over number rings, a fundamental problem at the intersection of number theory and logic.