J-invariantIn mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that Rational functions of j are modular, and in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of elliptic curves over C, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).
Local zeta functionIn number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as where V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements and Nm is the number of points of V defined over the finite field extension Fqm of Fq. Making the variable transformation u = q−s, gives as the formal power series in the variable .
Zero of a functionIn mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function , is a member of the domain of such that vanishes at ; that is, the function attains the value of 0 at , or equivalently, is the solution to the equation . A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function.
