p-adic modular forms

p-adic modular forms are mathematical objects that extend the classical theory of modular forms into the realm of p-adic numbers—an alternative number system emphasizing divisibility by a prime p. Unlike real or complex numbers, p-adic numbers focus on properties related to divisibility and convergence based on powers of p. p-adic modular forms allow mathematicians to study patterns and symmetries in number theory with tools suited for understanding congruences and local behaviors at p. They play a crucial role in modern number theory, especially in understanding deep connections between algebra, geometry, and arithmetic, such as in the proof of major conjectures like Fermat's Last Theorem.