On the Galois cohomology of ideal class groups

We use étale cohomology to prove some explicit results on the Galois cohomology of ideal class groups. Received: 3 May 2007     

Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5,7 or 10

We describe a way of constructing Jacobians of hyperelliptic curves of genus g ≥ 2, defined over a number field, whose Jacobians have a rational point of order of some (well chosen) integer l ≥ g + 1; the method is based on a polynomial identity. Using this approach we construct new families of genus 2 curves defined over — which contain the modular curves X₀(31) (and X₀(22) as a by-product) and X₀(29), the Jacobians of which have a rational point of order 5 and 7 respectively. We also construct a new family of hyperelliptic genus 3 curves defined over —, which contains the modular curve X₀(41), the Jacobians of which have a rational point of order 10. Finally we show that all hyperelliptic modular curves X₀(N) with N a prime number fit into the described strategy, except for N = 37 in which case we give another explanation. The authors thank the FNR (project FNR/04/MA6/11) for their support.     

Endomorphisms of Jacobians of modular curves

Let be the modular curve associated to a congruence subgroup Γ of level N with , and let be its canonical model over . The main aim of this paper is to show that the endomorphism algebra of its Jacobian is generated by the Hecke operators T _p , with , together with the “degeneracy operators” D _M,d , D ^t _M,d , for . This uses the fundamental results of Ribet on the structure of together with a basic result on the classification of the irreducible modules of the algebra generated by these operators. Received: 18 December 2007     

Notes on the minimal number of ramified primes in some l-extensions of Q

For a finite l-group G, let ram ^t (G) denote the minimal integer such that G can be realized as the Galois group of a tamely ramified extension of Q ramified only at ram ^t (G) finite primes. We study the upper bound of ram ^t (G) and give an improvement of the result of Plans. We also give the best bound of ram ^t (G) for all 3-groups G of order less than or equal to 3⁵. Received: 14 September 2007     

A five-dimensional modular variety

In this note we in the spirit of [1], we give ta structure theorem for a graded ring of modular forms related to the orthogonal group O(2, 5). Our results generalize some results obtained by Klöcker in his Ph.D thesis.     

A remark on roots of polynomials with positive coefficients

A short proof of a theorem of Dubickas on roots of polynomials with positive rational coefficients is presented.     

Determination of holomorphic modular forms by primitive Fourier coefficients

We prove that Siegel modular forms of degree greater than one, integral weight and level N, with respect to a Dirichlet character of conductor are uniquely determined by their Fourier coefficients indexed by matrices whose contents run over all divisors of . The cases of other major types of holomorphic modular forms are included. The author is supported by the Grant-in-Aid for JSPS fellows.     

On hyperelliptic modular curves over function fields

We prove that there are only finitely many modular curves of -elliptic sheaves over which are hyperelliptic. In odd characteristic we give a complete classification of such curves. The author was supported in part by NSF grant DMS-0801208 and Humboldt Research Fellowship.     

A new family of maximal curves over a finite field

It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type and defined over the finite field all have the maximum number of -rational points allowed by the Weil “explicit formulas”, and that these curves are -maximal curves over infinitely many algebraic extensions of . Serre showed that an -rational curve which is -covered by an -maximal curve is also -maximal. This has posed the problem of the existence of -maximal curves other than the Deligne–Lusztig curves and their -subcovers, see for instance Garcia (On curves with many rational points over finite fields. In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin, 2002) and Garcia and Stichtenoth (A maximal curve which is not a Galois subcover of the Hermitan curve. Bull. Braz. Math. Soc. (N.S.) 37, 139–152, 2006). In this paper, a positive answer to this problem is obtained. For every q = n ³ with n = p ^r  > 2, p ≥ 2 prime, we give a simple, explicit construction of an -maximal curve that is not -covered by any -maximal Deligne–Lusztig curve. Furthermore, the -automorphism group Aut has size n ³(n ³ + 1)(n ² − 1)(n ² − n + 1). Interestingly, has a very large -automorphism group with respect to its genus . Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni, PRIN 2006–2007.     

Connected but not path-connected subspaces of infinite graphs

Solving a problem of Diestel [9] relevant to the theory of cycle spaces of infinite graphs, we show that the Freudenthal compactification of a locally finite graph can have connected subsets that are not path-connected. However we prove that connectedness and path-connectedness to coincide for all but a few sets, which have a complicated structure.     

Pólya and Newtonian function fields

Let be a finite function field extension and denote by O _K the integral closure of in K. In this article, we are interested in Pólya fields, that is, fields K, such that the O _K -module Int(O _K ) of integer-valued polynomials over O _K admits a regular basis. We show that the cyclotomic extensions of are Pólya fields, and we characterize some totally imaginary extensions which are Pólya fields. Then, we are interested in Pólya fields K which have a regular basis of the form for some sequences of elements of O _K . For totally imaginary extensions, we show that it is the case if and only if O _K is isomorphic to . This gives a answer to a question raised by Thakur. The author thanks his thesis adviser Jean-Luc Chabert, and Mireille Car for their help, and their valuable advices to do this work. The author thanks also the referee for his valuable remarks.     

Coloring even-faced graphs in the torus and the Klein bottle

We prove that a triangle-free graph drawn in the torus with all faces bounded by even walks is 3-colorable if and only if it has no subgraph isomorphic to the Cayley graph C(Z ₁₃; 1,5). We also prove that a non-bipartite quadrangulation of the Klein bottle is 3-colorable if and only if it has no non-contractible separating cycle of length at most four and no odd walk homotopic to a non-contractible two-sided simple closed curve. These results settle a conjecture of Thomassen and two conjectures of Archdeacon, Hutchinson, Nakamoto, Negami and Ota. Institute for Theoretical Computer Science is supported as project 1M0545 by the Ministry of Education of the Czech Republic. The author was visiting Georgia Institute of Technology as a Fulbright scholar in the academic year 2005/06. Partially supported by NSF Grants No. DMS-0200595 and DMS-0354742.     

The product of two von Neumann n-frames, its characteristic, and modular fractal lattices

Let L be a bounded lattice. If for each a₁ < b₁ ∈ L and a₂ < b₂ ∈ L there is a lattice embedding ψ: [a₁, b₁] → [a₂, b₂] with ψ(a₁) = a₂ and ψ(b₁) = b₂, then we say that L is a quasifractal. If ψ can always be chosen to be an isomorphism or, equivalently, if L is isomorphic to each of its nontrivial intervals, then L will be called a fractal lattice. For a ring R with 1 let denote the lattice variety generated by the submodule lattices of R-modules. Varieties of this kind are completely described in [16]. The prime field of characteristic p will be denoted by F_p. Let be a lattice variety generated by a nondistributive modular quasifractal. The main theorem says that is neither too small nor too large in the following sense: there is a unique , a prime number or zero, such that and for any n ≥ 3 and any nontrivial (normalized von Neumann) n-frame of any lattice in , is of characteristic p. We do not know if in general; however we point out that, for any ring R with 1, implies . It will not be hard to show that is Arguesian. The main theorem does have a content, for it has been shown in [2] that each of the is generated by a single fractal lattice L_p; moreover we can stipulate either that L_p is a continuous geometry or that L_p is countable. The proof of the main theorem is based on the following result of the present paper: if is a nontrivial m-frame and is an n-frame of a modular lattice L with m, n ≥ 3 such that and , then these two frames have the same characteristic and, in addition, they determine a nontrivial mn-frame of the same characteristic in a canonical way, which we call the product frame. Presented by E. T. Schmidt.     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999     

Kummer's lemma for cyclic 2-groups

For cyclic 2-groupsC, we characterize the kernel of the map induced on the units of the integral group ring by the coefficient reduction . This allows us to prove that, for any finite abelian 2-groupA, the circular units of ℤA (i.e. those which are mapped to cyclotomic units by every character ofA) can be generated in a certain systematic way. Work supported in part by an NSERC (Canada) operating grant     

Genus formula for modular curves of {\mathcal{D}}-elliptic sheaves

We prove a genus formula for modular curves of -elliptic sheaves. We use this formula to show that the reductions of modular curves of -elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of tends to infinity. Received: 14 October 2008 The author was supported in part by NSF grant DMS-0801208 and Humboldt Research Fellowship.     

Congruences between Siegel modular forms

Using the moduli theory of abelian varieties and a recent result of Böcherer-Nagaoka on lifting of the generalized Hasse invariant, we show congruences between the weights of Siegel modular forms with congruent Fourier expansions. This result implies that the weights of p-adic Siegel modular forms are well defined.     

On the classification of Galois objects over the quantum group of a nondegenerate bilinear form

We study Galois and bi-Galois objects over the quantum group of a nondegenerate bilinear form, including the quantum group (SL(2)). We obtain the classification of these objects up to isomorphism and some partial results for their classification up to homotopy.     

Quotients of values of the Dedekind Eta function

Inspired by Riemann’s work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form , where A _j-1 and A _j are matrices whose rows are the coordinates of consecutive visible lattice points in a dilation XΩ of a fixed region Ω in , and z is a fixed complex number in the upper half plane. In particular, we show that the limiting distribution of these quotients depends heavily on the index of Farey fractions which was first introduced and studied by Hall and Shiu. The distribution of Farey fractions with respect to the value of the index dictates the universal limiting behavior of these quotients. Motivated by chains of these quotients, we show how to obtain a generalization, due to Zagier, of an important formula of Hall and Shiu on the sum of the index of Farey fractions. A. Zaharescu is supported by National Science Foundation Grant DMS-0456615.