For a curve of genus > 1 defined over a finite field, we present a sufficient criterion for the non-existence of automorphisms of order a power of a rational prime. We show how this criterion can be used to determine the automorphism group of some modular curves of high genus.
We study the logarithm of the least common multiple of the sequence of integers given by 12 + 1, 2 2 + 1, . . . , n2 + 1. Using a result of Homma  on the distribution of roots of quadratic polynomials modulo primes we calculate the error term for the asymptotics obtained by Cilleruelo
We present an algorithm for computing discriminants and prime ideal decomposition
in number fields. The algorithm is a refinement of a p-adic factorization method
based on Newton polygons of higher order. The running-time and memory requirements
of the algorithm appear to be very good: for a given prime number p, it computes the
p-valuation of the discriminant and the factorization of p in a number field of degree 1000
in a few seconds, in a personal computer.
For any Eichler order O(D, N) of level N in an indefinite
quaternion algebra of discriminant D there is a Fuchsian
group G(D, N) ¿ SL(2, R) and a Shimura curve X(D, N). We
associate to O(D, N) a set H(O(D, N)) of binary quadratic forms
which have semi-integer quadratic coefficients, and we develop a
classification theory, with respect to G(D, N), for primitive forms
contained in H(O(D, N)). In particular, the classification theory
of primitive integral binary quadratic forms by SL(2,Z) is
recovered. Explicit fundamental domains for G(D, N) allow the
characterization of the G(D, N)-reduced forms.