Varieties via their L-functions

We describe a procedure for determining the existence, or non-existence, of an algebraic variety of a given conductor via an analytic calculation involving L-functions. The procedure assumes that the Hasse-Weil L-function of the variety satisfies its conjectured functional equation, but there is no assumption of an associated automorphic object or Galois representation. We demonstrate the method by finding the Hasse-Weil L-functions of all hyperelliptic curves of conductor less than 500.Comment: 14 pages, 2 figure

The highest lowest zero of general <i>L</i>-functions

Stephen D. Miller showed that, assuming the Generalized Riemann Hypothesis, every entire L-function of real archimedean type has a zero in the interval 12+it with -t0<t<t0, where t0≈14.13 corresponds to the first zero of the Riemann zeta function. We give a numerical example of a self-dual degree-4 L-function whose first positive imaginary zero is at t1≈14.496. In particular, Miller's result does not hold for general L-functions. We show that all L-functions satisfying some additional (conjecturally true) conditions have a zero in the interval (-t2, t2) with t2≈22.661

Maass Forms on GL(3) and GL(4)

We describe a practical method for finding an L-function without first finding an associated underlying object. The procedure involves using the Euler product, the Ramanujan bound, and the approximate functional equation in a new way. No use is made of the functional equation of twists of the L-function. The method is used to find a large number of Maass forms on (SL 3,Z) and to give the first examples of Maass forms of higher level on GL(3), and on GL(4) and Sp(4)