There exist bialgebras with a left antipode but no right antipode(J.A.Green,W.D.Nichols,E.J.Taft,J.Algebra 65,399-411). We try toconstruct such a left Hopf algebra in the framework of quantum groups.We start with $3$ of the $6$ relations defining quantum $GL(2)$,plus inverting the quantum determinant. In asking that the left antipode, with itsstandard action on the $4$ generators, be an algebra antiendomorphism, weare forced to add new relations. The process stops at a Hopf algebra( two-sided) which seems to be new. It has the unusual feature that itremains non-commutative when $q=1$. Recently, we have dropped thecondition that the left antipode be an algebra antiendomorphism, but try to make it reverse the product only on irreducible words in thegenerators( there is a Birkhoff-Witt type basis). This almost works,but causes trouble on one nasty irreducible word. We hope to overcome this. ( Joint work with Suemi Rodriguez-Romo)

I will compare some examples of density of sets of primes in numberfields and function fields with those in graphs. The results require ArtinL-functions of Galois coverings of graphs.