L-series

L-series are mathematical functions that encode important information about objects called algebraic varieties, which are geometric shapes defined by polynomial equations. These functions, similar to complex functions like the Riemann zeta function, are constructed through infinite series or products and relate to the way these shapes behave over different number systems, especially finite fields. L-series help mathematicians study properties like symmetry, distribution of solutions, and deep connections between geometry and number theory, playing a central role in understanding fundamental mathematical structures and conjectures such as the Birch and Swinnerton-Dyer conjecture.