In this paper we study the following question: Is it true that a generic hypersurface X of degree d in P$^n+1$, where (d,n)≠ (3,1), (2,2), (3,2), does not admit a non-trivial, non-isomorphic surjective map to another smooth variety Y, except of course P$^n$? It is easy to see that it is true for n=1,2. We try to prove this for n=3 and can exclude all possibilities for Y except Y= G(1,4)∩ P$^6$and Y=V$_22^s$, a special Fano threefold of type V$_22$ found by Mukai and Umemura in [MU].

We describe a method for complete solution of the superelliptic Diophantine equation ay$^p$=f(x). The method is based on Baker‘s theory of linear forms in the logarithms. The characteristic feature of our approach (as compared with the classical method) is that we reduce the equation directly to the linear forms in logarithms, without intermediate use of Thue and linear unit equations. We show that the reduction method of Baker and Davenport [3] is applicable for superelliptic equations, and develop a very efficient method for enumerating the solutions below the reduced bound. The method requires computing the algebraic data in number fields of degree pn(n-1)/2 at most; in many cases this number can be reduced. Two examples with p=3 and n=4 are given.

To an arbitrary intersection of exceptional varieties of an embedded resolution we associate a finite number of congruences between naturally occurring multiplicities. This theory generalizes previous results concerning just one exceptional variety. Moreover we describe precise equalities which imply the congruences and we give some applications on the poles of Igusa‘s local zeta function.

Given an irreducible, modular, mod p representation p, we analyse the local components at p of newforms f which give rise to it.