Arithmetic and number theoretic techniques for the applied mathematician

This is a preliminary lecture that aims to give a broad overview into some of the concepts and tools that analytic number theorists and arithmetic geometers have at their disposal viz. modular forms and their generalizations, and L-functions attached to various objects. Their mathematical importance is difficult to be overstated. However, these tools, techniques and concepts have applications to a host of problems in mathematical physics too (not necessarily string theory). One such application is in the computation of certain Feynman integrals and scattering amplitudes.

The goal of this lecture is threefold:

1. The primary focus of this lecture will a very rudimentary and (hopefully) computational introduction to modular forms, L functions of elliptic curves and rational point counts. But the ulterior motive (pun intended) is to build towards a lecture series and/or seminar series, for which some have already expressed interest. More details to be discussed.
2. To introduce a few more dangerous words and related words to Bernd's list.
3. To build towards understanding (in an arithmetic setting) some of the dangerous words in Bernd's list.