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The number of rational points of a plane non-singular algebraic curve X defined over a finite field is computed, provided that the generic point of X is not an inflexion and that X is Frobenius non-classical with respect to conics.

Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups.…

Let k=F_q be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed…

We give a classification of rotational cmc surfaces in non-Euclidean space forms in terms of explicit parametrizations using Jacobi elliptic functions. Our method hinges on a Lie sphere geometric description of rotational linear Weingarten…

Superconformal surfaces in Euclidean space are the ones for which the ellipse of curvature at any point is a nondegenerate circle. They can be characterized as the surfaces for which a well-known pointwise inequality relating the intrinsic…

Regular model sets, describing the point positions of ideal quasicrystallographic tilings, are mathematical models of quasicrystals. An important result in mathematical diffraction theory of regular model sets, which are defined on locally…

By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof \`a la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an…

We construct spaces of 1-dimensional supersymmetric Euclidean field theories and show that they represent real or complex K-theory. A noteworthy feature of our bordism category is that the identity bordism of a point is connected to…

Merel's result on the strong uniform boundedness conjecture made it meaningful to classify the torsion part of the Mordell-Weil groups of all elliptic curves defined over number fields of fixed degree $d$. In this paper, we discuss the…

We classify the configurations of lines and conics in smooth Kummer quartics, assuming that all $16$ Kummer divisors map to conics. We show that the number of conics on such a quartic is at most $800$.

We establish a classification of cubic minimal cones in case of the so-called radial eigencubics. Our principal result states that any radial eigencubic is either a member of the infinite family of eigencubics of Clifford type, or belongs…

We prove that for all natural numbers $m$ and $k$ where $k$ is odd, there exists a natural number $N(k)$ such that any 3-connected cubic graph with at least $N(k)$ vertices contains a cycle of length $m$ modulo $k$. We also construct a…

We classify all cubic systems possessing the maximum number of invariant straight lines (real or complex) taking into account their multiplicities. We prove that there are exactly 23 topological different classes of such systems. For every…

In this short note we try to generalize the Clemens-Griffiths criterion of non-rationality for smooth cubic threefolds to the case of smooth cubic fourfolds.

We generalize the notion of cyclic codes by using generator polynomials in (non commutative) skew polynomial rings. Since skew polynomial rings are left and right euclidean, the obtained codes share most properties of cyclic codes. Since…

Using the Riemann Hypothesis over finite fields and bounds for the size of spherical codes, we give explicit upper bounds, of polynomial size with respect to the size of the field, for the number of geometric isomorphism classes of…

For an integer $k\geq 1$, a graph is called a $k$-circulant if its automorphism group contains a cyclic semiregular subgroup with $k$ orbits on the vertices. We show that, if $k$ is even, there exist infinitely many cubic arc-transitive…

We give a new representation of Euclidean quantum fields as scaling limits of systems of interacting, continuous, classical particles in the grand canonical ensemble.

We show that any rational cubic hypersurface of dimension at least 33 defined over a number field $K$ vanishes on a $K$-rational projective line, reducing the previous lower bound of Wooley by two. For $K=\mathbb Q$ we can reduce the bound…

A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal…