Serre Conjecture

Serre's Conjecture is a mathematical hypothesis about Galois representations, which encode symmetries of solutions to polynomial equations. It suggests that certain types of these representations, specifically those that are mod p (reducing coefficients modulo a prime p), are directly related to modular forms—special functions with symmetrical properties. In essence, the conjecture proposes a deep link between algebraic objects describing symmetries and analytical objects with rich transformation behaviors, helping mathematicians understand connections between number theory and complex analysis. It was a foundational step in modern number theory and was eventually proven by mathematician Chandrashekhar Khare and Jean-Pierre Wintenberger.