Principal Divisors Research Articles

On homeomorphisms preserving principal divisors

ABSTRACr. Let SI and S2 be compact Riemann surfaces of genus g > 1. Let T: SI S2 be a continuous map. The map T induces a group homomorphism from the group of divisors on S, into the group of divisors on S2* This group homomorphism will be denoted by the same letter T throughout this paper. If D = I mjp is a divisor on SI, then T(D)= XnI miT(p). If T is a holomorphic or an anti-holomorphic homeomorphism, then T(D) is a principal divisor on S2 if D is a principal divisor on SI. To what extent is the converse of this statement true? The answer to this question is provided by Theorem I of this paper: If T(D) is a principal divisor on S2 whenever D is a principal divisor on SI, then T is either a holomorphic or an anti-holomorphic homeomorphism.

A further note on principal divisors in arithmetic progressions

Assume that the class number h of an algebraic number field k is relatively prime to ( k : Q) and that k Q is galois. If ( a, m) = 1, then there are infinitely many primes p≡a mod m which have principal prime ideal factors in k.

On a problem of Schinzel concerning principal divisors in arithmetic progressions

On a problem of Schinzel concerning principal divisors in arithmetic progressions

Euclid's Algorithm in Algebraic Function Fields

It has been pointed out to me by Dr. J. W. S. Cassels† that ∂7 of the above paper rests on an ambigious use of the term “ class number of K ”. The class number of the Riemann-Roch Theorem is the order of the group of rational divisors of degree 0 modulo the principal divisors, whilst the class number which is relevant to the questions of unique factorization and of Euclid's Algorithm is the order of the group of ideal classes in the ring, I, of S-integers of K, where S denotes the set of places of K which lie above the place p1/x of the transcendental field k(x). We denote these class numbers by h and hx respectively. In order to obtain a connection between the two, we introduce the concept of the regulator, rx, of the group of ideal classes of I. Let D∗ denote the group of principal divisors of K and let U0 denote the group of divisors of K based on S and of degree 0. Then rx is defined to be the index of D∗ ∩ U0 in U0. Finally, we denote by sx the greatest common divisor of the degrees of the places in S. One can show that Sxh = rxhx (1) ( cf. [1; pp. 31, 32]).

Survey Article-Diophantine Equations with Special Reference to Elliptic Curves

(I). The middle line of equation (6.3) on p. 210 should read : “ =0 or 1 if deg D = 0 ”. The D with l(D) = 1 and deg D = 0 are, of course, precisely the principal divisors. (II). There is an unfortunate confusion in the first six lines of p. 278. To prove Šafarevič's theorem one must consider not IIIm but цm, where ц is given by (27.1). Since this group is also finite, the argument works with this alteration. The finiteness of цm for every m follows at once from the finiteness of цq for every prime q. And this is a consequence of Theorem 7.1 of Cassels (1962a) [cf. also the reformulation on p. 153 of Cassels (1964b)] combined with the finiteness of the Selmer group S(q). (III). Minor misprints. In the second footnote on p. 254 replace the first “ = ” by “ ≠ ”. On p. 284, line 7 replace “ g' ” by “ g ” and the second “ ½ ” by “ 0 ”. (IV). The references Serre (1964b) and Tate (1964a) are now generally available in English in: Arithmetic algebraic geometry, Proceedings of a conference held at Purdue University, December 5–7, 1963 (Harper and Row, New York, 1966).

Extremal length of weak homology classes on Riemann surfaces

1. The square integrable harmonic differentials on a Riemann surface W form a Hilbert space rh. Let rP be a closed subspace of rh. Let c be a 1-chain on W. There exists a unique element i1(c) CrP with the property flco= (c, {1'(c)) for all coCPr. We refer to it(c) as the F.reproducing differential for c. Accola [1] has shown that if c is a cycle, then the extremal length of the homology class of c is equal to the square of the norm of the rh-reproducing differential for c (cf. also [3]). Two specific problems raised by Accola's result are the following. For the important subspaces rF, does the norm of the rFreproducing differential for a cycle have an extremal length interpretation? Secondly, we may ask for a family of curves associated with a 1-chain c, not necessarily a cycle, whose extremal length gives the norm of the rh-reproducer for c. (By Abel's theorem, the vanishing of the norm of this reproducer implies that ac is a principal divisor.) In the present paper we give an answer to the first question for the subspace rh8e. Theorem 1 states that an associated geometric configuration is the weak homology class of c.2