Torsion points on the jacobian of a Fermat quotient

Let p≥5 be a prime, ζ a primitive pth root of unity and λ=1−ζ. For 1≤s≤p−2, the smooth projective model C_p,s of the affine curve v^p=u^s(1−u) is a curve of genus (p−1)/2 whose jacobian J_p,s has complex multiplication by the ring of integers of the cyclotomic field Q(ζ). In 1981, Greenberg determined the field of rationality of the p-torsion subgroup of J_p,s and moreover he proved that the λ³-torsion points of J_p,s are all rational over Q(ζ). In this paper we determine quite explicitly the λ³-torsion points of J_p,1 for p=5 and p=7, as well as some further p-torsion points which have interesting arithmetical applications, notably to the complementary laws of Kummer’s reciprocity for pth powers.     

Densité de points et minoration de hauteur

We obtain a lower bound for the normalised height of a non-torsion subvariety V of a C.M. abelian variety. This lower bound is optimal in terms of the geometric degree of V, up to a power of a “log”. We thus extend the results of Amoroso and David on the same problem on a multiplicative group . We prove furthermore that the optimal lower bound (conjectured by David and Philippon) is a corollary of the conjecture of David and Hindry on the abelian Lehmer's problem. We deduce these results from a density theorem on the non-torsion points of V.     

Torsion points on families of products of elliptic curves

In a recent paper we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes. Here we go ahead with the programme of settling the conjecture for general abelian surface schemes by completing the proof for all non-simple surfaces. This involves some entirely new and crucial issues.     

Periods of orbits modulo primes

Let S be a monoid of endomorphisms of a quasiprojective variety V defined over a global field K. We prove a lower bound for the size of the reduction modulo places of K of the orbit of any point α∈V(K) under the action of the endomorphisms from S. We also prove a similar result in the context of Drinfeld modules. Our results may be considered as dynamical variants of Artin's primitive root conjecture.     

Orbits of points on certain K3 surfaces

In this paper we show that, for a K3 surface within a certain class of surfaces and over a number field, the orbit of a point under the group of automorphisms is either finite or its exponent of growth is exactly the Hausdorff dimension of a fractal associated to the ample cone. In particular, the exponent depends on the geometry of the surface and not its arithmetic. For surfaces in this class, the exponent is 0.6527±0.0012.     

The <Emphasis Type="Italic">L</Emphasis>-series of a cubic fourfold

We study the L-series of cubic fourfolds. Our main result is that, if X/C is a special cubic fourfold associated to some polarized K3 surface S, defined over a number field K and satisfying , then X has a model over K such that the L-series of the primitive cohomology of X/K can be expressed in terms of the L-series of S/K. This allows us to compute the L-series for a discrete dense subset of cubic fourfolds in the moduli spaces of certain special cubic fourfolds. We also discuss a concrete example.     

Disguised toric dynamical systems

We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric. They are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. By means of (real) algebraic geometry we show that some reaction networks have an empty toric locus or a toric locus of Lebesgue measure zero in parameter space, while their disguised toric locus is of positive measure. We also propose some algorithms one can use to detect the disguised toric locus.     

Cubic surfaces violating the Hasse principle are Zariski dense in the moduli scheme

We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic surfaces.     

A Chebyshev spectral collocation method for solving Burgers’-type equations

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge–Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers’, KdV–Burgers’, coupled Burgers’, 2D Burgers’ and system of 2D Burgers’ equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds’ number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.     

Rational points of bounded height on projective surfaces

We give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces.     

Rational points and cohomology of discriminant varieties

Let G be a finite unitary reflection group acting in a complex vector space . The discriminant varietyX_G of G is defined as the space of regular orbits of G on V. Classical examples include the varieties of complex polynomials of degree n with distinct (resp. non-zero distinct) roots. The normaliser of G in GL(V) acts on X_G; in this work we determine the action of on the cohomology of X_G. In the classical cases this amounts to computing the cohomology of X_G with certain local coefficient systems. Our methods are to compute equivariant weight polynomials by means of explicit counting of the rational points of certain varieties over finite fields, and then to exploit the weight purity of the relevant varieties. We obtain some power series identities as a byproduct.     

A note on the representation of positive polynomials with structured sparsity

We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain way. Let the solution set of a finite system of polynomial inequalities be given, where each inequality involves only variables of one block. We investigate polynomials that are positive on such a set and sparse in the sense that each monomial involves only variables of one block. In particular, we derive a short and direct proof for Lasserre’s theorem on the existence of sums of squares certificates respecting the block structure. The motivation for the results can be found in the literature on numerical methods for global optimization of polynomials that exploit sparsity. The first and the third author were supported by the DFG grant “Barrieren”. The second author was supported by “Studienstiftung des deutschen Volkes”.     

Periodic non-autonomous second-order dynamical systems

We study the existence of periodic solutions for a second-order non-autonomous dynamical system. We give three sets of hypotheses which guarantee the existence of non-constant solutions. We were able to weaken the hypotheses considerably from those used previously for such systems. We employ a saddle point theorem using linking methods.     

Periodic points classify a family of Markov shifts

Ledrappier introduced the following type of space of doubly indexed sequences over a finite abelian group G,

     

Dyson's theorem for curves

Let K be a number field and X₁ and X₂ two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson theorem for the product X₁×X₂. If Xi=P₁ we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on integral points on hyperbolic curves and it will give some insight on effectiveness. This proof is new and avoids the use of Roth and Mordell-Weil theorems, the theory of Linear Forms in Logarithms and the Schmidt subspace theorem.     

Ring structures on groups of arithmetic functions

Using the theory of Witt vectors, we define ring structures on several well-known groups of arithmetic functions, which in another guise are formal Dirichlet series. The set of multiplicative arithmetic functions over a commutative ring R is shown to have a unique functorial ring structure for which the operation of addition is Dirichlet convolution and the operation of multiplication restricted to the completely multiplicative functions coincides with point-wise multiplication. The group of additive arithmetic functions over R also has a functorial ring structure. In analogy with the ghost homomorphism of Witt vectors, there is a functorial ring homomorphism from the ring of multiplicative functions to the ring of additive functions that is an isomorphism if R is a Q-algebra. The group of rational arithmetic functions, that is, the group generated by the completely multiplicative functions, forms a subring of the ring of multiplicative functions. The latter ring has the structure of a Bin(R)-algebra, where Bin(R) is the universal binomial ring equipped with a ring homomorphism to R. We use this algebra structure to study the order of a rational arithmetic function, as well the powersfα for α∈Bin(R) of a multiplicative arithmetic function f. For example, we prove new results about the powers of a given multiplicative arithmetic function that are rational. Finally, we apply our theory to the study of the zeta function of a scheme of finite type over Z.     

Estimates for Tamagawa numbers of diagonal cubic surfaces

For diagonal cubic surfaces, we give an upper bound for E. Peyre's Tamagawa type number in terms of the coefficients of the defining equation. This bound shows that the reciprocal admits a fundamental finiteness property on the set of all diagonal cubic surfaces. As an application, we show that the infinite series of Tamagawa numbers related to the Fano cubic bundles considered by Batyrev and Tschinkel (1996) [BT] are indeed convergent.     

Asymptotic behavior of coupled dynamical systems with multiscale aspects

We study the asymptotic behavior, as time variable t goes to +∞, of nonautonomous dynamical systems involving multiscale features. As a benchmark case, given H a general Hilbert space, and two closed convex functions, and β a function of t which tends to +∞ as t goes to +∞, we consider the differential inclusion

     

Weierstrass weight of the hyperosculating points of generalized Fermat curves

Let $(S,H)$ be a generalized Fermat pair of the type $(k,n)$. If $F?S$ is the set of fixed points of the non-trivial elements of the group H, then F is exactly the set of hyperosculating points of the standard embedding $S?{\mathbb{P}}^n$. We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.