Büchi's problem asked whether there exists an integer $M$ such that the surface defined by a system of equations of the form

$$x_{n}^2+x_{n-2}^2=2x_{n-1}^2+2,\quad n=2,\dotsc, M-1,$$

has no integer points other than those that satisfy $\pm x_n=\pm x_0+n$ (the $\pm$ signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system $Q=(q_1,\dotsc,q_r)$ of integral quadratic forms and an arbitrary $r$-tuple $B=(b_1,\dotsc,b_r)$ of integers, whether $Q$ represents $B$ (see T. Pheidas and X. Vidaux, Fund. Math. 185 (2005) 171–194). Thus it would imply the following strengthening of the negative answer to Hilbert's tenth problem: the positive-existential theory of the rational integers in the language of addition and a predicate for the property ‘$x$ is a square’ would be undecidable. Despite some progress, including a conditional positive answer (depending on conjectures of Lang), Büchi's problem remains open.

In this paper we prove the following:

1. an analogue of Büchi's problem in rings of polynomials of characteristic either 0 or $p\geq17$ and for fields of rational functions of characteristic 0; and

2. an analogue of Büchi's problem in fields of rational functions of characteristic $p\geq19$, but only for sequences that satisfy a certain additional hypothesis.

Let $F$ be a field of characteristic either 0 or at least 17 and let $t$ be a variable. Let $L_{t}$ be the first order language which contains symbols for 0 and 1, a symbol for addition, a symbol for the property ‘$x$ is a square’ and symbols for multiplication by each element of the image of $\mathbb{Z}[t]$ in $F[t]$. Let $R$ be a subring of $F(t)$, containing the natural image of $\mathbb{Z}[t]$ in $F(t)$. Assume that one of the following is true:

1. $R\subset F[t]$;

2. the characteristic of $F$ is either 0 or $p\geq19$.

We determine which groups ${\mathbb{Z}}/M{\mathbb{Z}}\oplus{\mathbb{Z}}/N{\mathbb{Z}}$ occur infinitely often as torsion groups $E(K)_{\operatorname{tors}}$ when $K$ varies over all quartic number fields and $E$ varies over all elliptic curves over $K$.

Using properties of the Frobenius eigenvalues, we show that, in a precise sense, ‘most’ isomorphism classes of (principally polarized) simple abelian varieties over a finite field are characterized, up to isogeny, by the sequence of their division fields, and we show a similar result for ‘most’ isogeny classes. Some global cases are also treated.

It is proved that the finite soluble groups can be characterized among finite groups by a first-order sentence, namely, the sentence that asserts that no non-trivial element $g$ is a product of 56 commutators $[x,y]$ with entries $x$, $y$ conjugate to $g$.

We prove a lifting theorem (from any characteristic $p$ to characteristic zero) for holomorphic Picard modular forms of any weight and level. The proof makes use of the knowledge of the geometry of the toroidal compactification of the Picard modular surface.

Benford's law (to base $B$) for an infinite sequence $\{x_k: k \ge 1\}$ of positive quantities $x_k$ is the assertion that $\{ \log_B x_k : k \ge 1\}$ is uniformly distributed $(\bmod\ 1)$. The $3x+1$ function $T(n)$ is given by $T(n)=(3n+1)/{2}$ if $n$ is odd, and $T(n)= n/2$ if $n$ is even. This paper studies the initial iterates $x_k= T^{(k)}(x_0)$ for $1 \le k \le N$ of the $3x+1$ function, where $N$ is fixed. It shows that for most initial values $x_0$, such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence $\{\log_B x_k: 1 \le k \le N \}$ is small.

Smooth complete algebraic varieties defined over an algebraically closed field are divided into two classes according to the behaviour of the albanese map. Criteria are given for when a variety belongs to either class. The relationship with the Tate conjecture for abelian varieties is discussed.

Let $\mathbb K$ be an algebraically closed field, let $X$ be a $\mathbb K$-variety, and let $X(\mathbb K)$ be the set of closed points in $X$. A constructible set$C$ in $X(\mathbb K)$ is a finite union of subsets $Y(\mathbb K)$ for subvarieties $Y$ in $X$. A constructible function$f:X(\mathbb K)\rightarrow\mathbb Q$ has $f(X(\mathbb K))$ finite and $f^{-1}(c)$ constructible for all $c\ne 0$. Write CF$(X)$ for the vector space of such $f$. Let $\phi:X\rightarrow Y$ and $\psi: Y\rightarrow Z$ be morphisms of ${\mathbb C}$-varieties. MacPherson defined a linear pushforward CF$(\phi):{\rm CF}(X)\rightarrow{\rm CF}(Y)$ by ‘integration’ with respect to the topological Euler characteristic. It is functorial, that is, CF$(\psi\circ\phi)={\rm CF}(\psi)\circ{\rm CF}(\phi)$. This was extended to $\mathbb K$ of characteristic zero by Kennedy.

This paper generalizes these results to $\mathbb K$-schemes and Artin$\mathbb K$-stacks with affine stabilizer groups. We define the notions of Euler characteristic for constructible sets in $\mathbb K$-schemes and $\mathbb K$-stacks, and pushforwards and pullbacks of constructible functions, with functorial behaviour. Pushforwards and pullbacks commute in Cartesian squares. We also define pseudomorphisms, a generalization of morphisms well suited to constructible functions problems.

This paper presents a rigidity theorem for infinite-dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space $A^{1}(M)$, for such a Riemann surface $M$, is isomorphic to the Banach space of summable sequence, $l^{1}$. This implies that whenever $M$ and $N$ are Riemann surfaces that are not analytically finite, and in particular are not necessarily homeomorphic, then $A^{1}(M)$ is isomorphic to $A^{1}(N)$. It is known from V. Markovic that if there is a linear isometry between $A^{1}(M)$ and $A^{1}(N)$, for two Riemann surfaces $M$ and $N$ of non-exceptional type, then this isometry is induced by a conformal mapping between $M$ and $N$. As a corollary to this rigidity theorem presented here, taking the Banach duals of $A^{1}(M)$ and $l^{1}$ shows that the space of holomorphic quadratic differentials on $M,\ Q(M)$, is isomorphic to the Banach space of bounded sequences, $l^{\infty }$. As a consequence of this theorem and the Bers embedding, the Teichmüller spaces of such Riemann surfaces are locally bi-Lipschitz equivalent.

We show the existence of surfaces of degree $d$ in ${\mathbb P}^3({\mathbb C})$ with approximately $(3j+2)/(6j(j+1))\,d^3$ singularities of type $A_j, 2\le j\le d-1$. The result is based on Chmutov's construction of nodal surfaces. For the proof we use plane trees related to the theory of Dessins d'Enfants.

Our examples improve the previously known lower bounds for the maximum number $\mu_{A_j}(d)$ of $A_j$-singularities on a surface of degree $d$ in most cases. We also give a generalization to higher dimensions which leads to new lower bounds even in the case of nodal hypersurfaces in ${\mathbb P}^n, n\geq5$.

To conclude, we work out in detail a classical idea of Segre which leads to some interesting examples, for example, to a sextic with 36 cusps.

We consider the classical universal covering $\exp: {{\mathbb C}}\To {{\mathbb C}}^*$ of the complex torus as an algebraic structure. The exponentiation is seen here as an abstract homomorphism from a divisible torsion-free group onto the multiplicative group of an algebraically closed field of characteristic zero, with cyclic kernel. We prove that any structure satisfying this description is isomorphic to the classical one provided that the cardinality of the underlying field is equal to that of ${{\mathbb C}}$. This can also be seen as a model-theoretic statement on the categoricity of a corresponding $L_{\omega_1,\omega}$-sentence. The proof is a combination of arithmetic and model-theoretic methods.

We prove a general theorem showing that iterated skew polynomial extensions of the type that fit the conditions needed by Cauchon's deleting derivations theory and by the Goodearl–Letzter stratification theory are unique factorisation rings in the sense of Chatters and Jordan. This general result applies to many quantum algebras; in particular, generic quantum matrices and quantized enveloping algebras of the nilpotent part of a semisimple Lie algebra are unique factorisation domains in the sense of Chatters. The result also extends to generic quantum grassmannians (by using noncommutative dehomogenisation) and to the quantum groups ${\mathcal O}_q(\text{GL}_n)$ and ${\mathcal O}_q(\text{SL}_n)$.

Let $N$ be a point of an orbit closure $\overline{{\mathcal O}}_M$ in a module variety such that its orbit ${\mathcal O}_N$ has codimension 2 in $\overline{{\mathcal O}}_M$. We show that under some additional conditions the pointed variety $(\overline{{\mathcal O}}_M,N)$ is smoothly equivalent to a cone over a rational normal curve.

Let $Y$ be a reduced irreducible projective curve of arithmetic genus $g\ge 2$ with at most ordinary nodes as singularities. For a subsheaf $F$ of rank $r'$, degree $d'$ of a torsion-free sheaf $E$ of rank $r$, degree $d$, let $s(E,F) = r'd-rd'$. Define $s_{r'}(E) = {\rm min} s(E,F)$, where the minimum is taken over all subsheaves of $E$ of rank $r'$. For a fixed $r', s_{r'}$ defines a stratification of the moduli space $U(r,d)$ of stable torsion-free sheaves of rank $r$, degree $d$ by locally closed subsets $U_{r',s}$. We study the nonemptiness and dimensions of the strata. We show that the general element in each nonempty stratum is a vector bundle and it has only finitely many (respectively unique) maximal subsheaves of rank $r'$ for $s\le r'(r-r')(g-1)$ (respectively $s<r'(r-r')(g-1)$). We prove that the tensor product of two general stable vector bundles on an irreducible nodal curve $Y$ is nonspecial.

In this paper we construct new obstructions for the surjectivity of the Johnson homomorphism of the automorphism group of a free group. We also determine the structure of the cokernel of the Johnson homomorphism for degrees 2 and 3.

We revise the concept of compact tripotent in the bidual space of a JB*-triple. This concept was introduced by Edwards and Rüttimann generalizing the ideas developed by Akemann for compact projections in the bidual of a C*-algebra. We also obtain some characterizations of weak compactness in the dual space of a JC*-triple, showing that a bounded subset in the dual space of a JC*-triple is relatively weakly compact if and only if its restriction to any abelian maximal subtriple $C$ is relatively weakly compact in the dual of $C$. This generalizes a very useful result by Pfitzner in the setting of C*-algebras. As a consequence we obtain a Dieudonné theorem for JC*-triples which generalizes the one obtained by Brooks, Saitô and Wright for C*-algebras.

Let $G$ be a finite (non-abelian) irreducible linear subgroup over the complex numbers, and let $g$ be an element of $G$ of prime order $p$. Suppose that $g$ does not belong to a proper normal subgroup of $G$. We show that the number of distinct eigenvalues of $g$ can only be $p,p-1,p-2,(p+1)/2$ or $(p-1)/2$. Moreover, we provide a full classification of such groups $G $ provided that $g$ has at most $p-2$ distinct eigenvalues.

Let $A$ be a graded algebra finitely generated in degree 1 over a field $k$. Point modules over $A$ introduced by Artin, Tate and Van den Bergh play an important role in studying $A$ in noncommutative algebraic geometry. In this paper, we define a dual notion of point module in terms of Koszul duality, which we call a co-point module. Using co-point modules, we will construct counter-examples to the following condition due to Auslander: for every finitely generated right module $\pi$ over a ring $R$, there is a natural number $n_M\in {\mathbb N}$ such that, for any finitely generated right module $N$ over $R$, ${\rm Ext}^i_R(M, N)=0$ for all $i\gg 0$ implies ${\rm Ext}^i_R(M, N)=0$ for all $i>n_M$.

We consider the cohomology of Mackey functors for infinite groups and define the Mackey-cohomological dimension ${\rm cd}_{{\mathfrak M}} G$ of a group $G$. We will relate this dimension to other cohomological dimensions such as the Bredon-cohomological dimension ${\rm cd}_{{\mathfrak F}}G$ and the relative cohomological dimension ${{\mathfrak F}}\text{-}{\rm cd} G$. In particular, we show that for virtually torsion free groups the Mackey-cohomological dimension is equal to both ${{\mathfrak F}}\text{-}{\rm cd} G$ and the virtual cohomological dimension.

On a large class of post-critically finite (finitely ramified) self-similar fractals with possibly little symmetry, we consider the question of existence and uniqueness of a Laplace operator. By considering positive refinement weights (local scaling factors) which are not necessarily equal, we show that for each such fractal, under a certain condition, there are corresponding refinement weights which support a unique self-similar Dirichlet form. As compared with previous results, our technique allows us to replace symmetry by connectivity arguments.