Computational Algebra Seminar
I will describe how L-functions can be attached to various number theoretic objects, and comment on the motivation for wanting to compute with these L-functions, in particular what types of conjectures they are supposed to obey. Then I will give some specific examples from work that has derived from an NSF Focused Research Group grant, particularly some experiments with rank 4 quadratic twists, and some appearances of eta-quotients with Waldspurger lifts of modular forms. With the latter, in some examples it is possible to write the q-series as a product of two weighted theta series, and then the use of FFT convolution can be used to get a large initial segment (up to 1010 in some cases).
Part 1: cones and fans.
Affine 2-space k2 (over a field k) has the polynomial ring k[x,y] as its ring of (polynomial) functions. If σ in ℝ2 is the first quadrant and M = ℤ2 is the integral lattice in ℝ2, then I can regard the set of all monomials xi yj (and their multiplicative structure) as σ∩M. Thus k[x,y] = k[σ∩M], and so I think of σ (sitting in M⊗ℚ) as determining k2.
Toric geometry makes a lot out of this by allowing other choices of cones than σ, allowing collections of cones, and considering maps between cones. Many features of geometry can be interpreted as questions about cones in lattices. I'll go through the famous first examples.