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There is a well-known formula for the Hurwitz zeta function which implies the functional equation for the Riemann zeta function. We give a new proof of that formula and recover certain integral representations for the Hurwitz and Riemann zeta functions.

The proof of Theorem 3.2 in [1] (P. Tzermias, On the p-adic binomial series and a formal analogue of Hilbert's Theorem 90, Glasgow Math. J.47 (2005), 319–326) contains two opaque claims. The necessary clarifications are provided here.

We prove that the torsion part of the Mordell–Weil group of the Jacobian of a Fermat curve over a cyclotomic field is contained in the kernel of a certain isogeny. This provides a natural analogue of a similar result on Jacobians of Fermat quotient curves.

We give an explicit version of a classical theorem of Stickelberger on the representation of certain integers by binary quadratic forms. This is achieved by generalizing Stickelberger's original congruences via an extension of a recent result of Young.

Let p be a prime greater than or equal to 17 and congruent to 2 modulo 3. We use results of Beukers and Helou on Cauchy–Liouville–Mirimanoff polynomials to show that the intersection of the Fermat curve of degree p with the line X + Y = Z in the projective plane contains no algebraic points of degree d with 3 ≤ d ≤ 11. We prove a result on the roots of these polynomials and show that, experimentally, they seem to satisfy the conditions of a mild extension of an irreducibility theorem of Pólya and Szegö. These conditions are conjecturally also necessary for irreducibility.

We investigate low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve. We compute all quadratic points on these curves and use Coleman’s effective Chabauty method to obtain bounds for the number of cubic points on each of the former two curves.

The author is a Corresponding Member of the Academy of Athens, University of Fribourg (Switzerland) (1965-1995), University of Zurich (1984-1992), former Director of the European Cultural Centre of Delphi, former Correspondent of Neue Zürcher Zeitung. Tzermias is the author of a great number of books and essays concerning the relations between history and geography. As the director of the European Cultural Centre of Delphi he organized an international scientific conference on architecture and landscape.

We describe the group generated by the Weierstrass points in the Jacobian of the curve X 4 +Y 4 +Z 4 +3 (X 2 Y 2 +X 2 Z 2 +Y 2 Z 2 )) = 0 This curve is the only curve of genus 3, apart from the fourth Fermat curve, possessing exactly twelve Weierstrass points.

The Manin–Mumford conjecture asserts that if K is a field of characteristic zero, C a smooth proper geometrically irreducible curve over K, and J the Jacobian of C, then for any embedding of C in J, the set C(K) ∩ J(K)tors is finite. Although the conjecture was proved by Raynaud in 1983, and several other proofs have appeared since, a number of natural questions remain open, notably concerning bounds on the size of the intersection and the complete determination of C(K) ∩ J(K)tors for special families of curves C. The first half of this survey paper presents the Manin–Mumford conjecture and related general results, while the second describes recent work mostly dealing with the above questions.

This paper is concerned with the arithmetic of curves of the form vp=us(1-u), where p is a prime with p ≥ 5 and s is an integer such that 1 ≤ s ≤ p-2. The Jacobians of these curves admit complex multiplication by a primitive p-th root of unity ζ. We find explicit rational functions on these curves whose divisors are p-multiples of divisors representing (1-ζ)2 - and (1-ζ)3-division points on the corresponding Jacobians. This also gives an effective version of a theorem of Greenberg.

The Coleman-Chabauty bound is an upper bound for the number of rational points on a curve of genus g ≥ 2 whose Jacobian has Mordell-Weil rank r less than g . The bound is given in terms of the genus of the curve and the number of F p -points on the reduced curve, for all primes p of good reduction such that p > 2 g . In this Note we show that the hypothesis on the Mordell-Weil rank is essential. We do so by exhibiting, for each prime p ≥ 5, an explicit family of curves of genus ( p − 1)/2 (and rank at least ( p − 1)/2) for which the bound in question does not hold. Our examples show that the difference between the number of rational points and the bound in question can in fact be linear in the genus. Under mild assumptions, our curves have rank at least twice their genus.

We determine all algebraic points of degree at most five over Q on the Fermat curve of degree seven.

Let J 5 J_{5} denote the Jacobian of the Fermat curve of exponent 5 and let K = Q ( ζ 5 ) K=Q(\zeta _{5}) . We compute the groups J 5 ( K ) J_{5}(K) , J 5 ( K + ) J_{5}(K^{+}) , J 5 ( Q ) J_{5}(Q) , where K + K^{+} is the unique quadratic subfield of K K . As an application, we present a new proof that there are no K K -rational points on the 5-th Fermat curve, except the so called “points at infinity".

We study the torsion in the Mordell-Weil group of the Jacobian of the Fermat curve of exponent p over the cyclotomic field obtained by adjoining a primitive p-th root of 1 to Q. We show that for all (except possibly one) proper subfields of this cyclotomic field, the torsion parts of the corresponding Mordell-Weil groups are elementary abelian p-groups.

In his paper ("Oevres Scientifiques, Collected Papers, Vol. III, pp. 329-342, Springer-Verlag, Berlin/New York, 1979), Well asserted (without proof) that the automorphism group of the Fermat hypersurface of exponent N and dimension r − 1 over an algebraically closed field of characteristic prime to N is the semidirect product of the symmetric group on r + 1 letters and the direct sum of r copies of the cyclic group of order N. It turns out that the assertion is false in positive characteristic. In this paper, we present a proof of Weil′s assertion for the case of the Fermat curves (r = 2) in characteristic 0.

The main result of [23] on the non-existence of low-degree irreducible factors of the Cauchy-Liouville-Mirimanoff polynomials $E_p(x)$ for primes $p \equiv 2 \pmod{3}$ is extended to a similar result for $p \equiv 1 \pmod{3}$. We also give a partial result on the existence of higher-degree irreducible factors.