Zeros of families of automorphic L-functions close to 1

Zeros of Families of Automorphic L-functions Close to 1

For many L-functions of arithmetic interest, the values on or close to the edge of the region of absolute convergence are of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to non-vanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet L-functions (e.g. because of the Dirichlet class-number formula) and the symmetric square L-functions of ...

Zeros of Families of Automorphic L-functions

For many L-functions of arithmetic interest, the values on or close to the edge of the region of absolute convergence are of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to non-vanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet L-functions (e.g., because of the Dirichlet class-number formula) and the symmetric square L-functions of...

1 9 Ja n 19 99 Low Lying Zeros of Families of L - Functions

THE ANALYTIC RANK OF J0(q) AND ZEROS OF AUTOMORPHIC L-FUNCTIONS

We study zeros of the L-functions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the L-functions at the central critical point s = 1 2 is absolutely bounded. On the Birch and Swinnerton-Dyer conjecture, this provides an upper bound for the rank of the Jacobian J0(q) of the modular curve X0(q), which is of the sam...

Nonvanishing of Gl ( 2 ) automorphic L functions at 1 / 2

Let k be a number field with ring of adeles A, let B be a quaternion algebra defined over k, and letG = B×. Let π be an infinite dimensional irreducible cuspidal automorphic representation of G(A). Then the vanishing or nonvanishing of L(1/2, π) has been conjectured or shown to be equivalent to conditions of considerable interest in number theory or automorphic representation theory. For exampl...