Fermat Surface Research Articles

Fermat surfaces and hypercubes

<p>Fermat’s last theorem appears not as a unique property of natural numbers but as the bottom line of extended possible issues involving larger dimensions and powers when observed from a natural vector space viewpoint. The fabric of this general Fermat’s theorem structure consists of a well-defined set of vectors associated with <strong> </strong>dimensional vector spaces and the Minkowski norms one can define there. Here, a special vector set is studied and named a Fermat surface. Besides, a connection between Fermat surfaces and hypercubes is unveiled.</p>

On the supersingular locus of the Shimura variety for GU(2,2) over a ramified prime

We study the structure of the supersingular locus of the Rapoport–Zink integral model of the Shimura variety for [Formula: see text] over a ramified odd prime with the special maximal parahoric level. We prove that the supersingular locus equals the disjoint union of two basic loci, one of which is contained in the flat locus, and the other is not. We also describe explicitly the structure of the basic loci. More precisely, the former one is purely [Formula: see text]-dimensional, and each irreducible component is birational to the Fermat surface. On the other hand, the latter one is purely [Formula: see text]-dimensional, and each irreducible component is birational to the projective line.

On supersingular loci of Shimura varieties for quaternionic unitary groups of degree 2

We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree 2 over a ramified odd prime p if the level at p is given by a special maximal compact open subgroup. More precisely, we show that such a locus is purely 2-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport–Zink space for the quaternionic unitary similitude group of degree 2, with a special parahoric level. We prove that such a scheme is purely 2-dimensional, and every irreducible component is isomorphic to the Fermat surface. We also determine its connected components, irreducible components and their intersection behaviors by means of the Bruhat–Tits building of \({{\,\mathrm{PGSp}\,}}_4({\mathbb {Q}}_p)\). In addition, we compute the intersection multiplicity of the GGP cycles associated to an embedding of the considering Rapoport–Zink space into the Rapoport–Zink space for the unramified \({{\,\mathrm{GU}\,}}_{2,2}\) with hyperspecial level for the minuscule case.

Special components of Noether–Lefschetz loci

We take a sum $$C_1+rC_2, r\in {\mathbb {Q}}$$ of a line $$C_1$$ and a complete intersection curve $$C_2$$ of type (3, 3) inside a smooth surface of degree 8 and with $$C_1\cap C_2=\emptyset $$ . We gather evidences to the fact that for all except a finite number of $$r$$ , the Noether–Lefschetz loci attached to the cohomology classes of $$C_1+rC_2$$ are distinct 31 codimensional subvarieties intersecting each other in a 32 codimensional subvariety of the ambient space. The maximum codimension for components of the Noether–Lefschetz locus in this case is 35, and hence, we provide a conjectural description of a counterexample to a conjecture of J. Harris. The methods used in this paper also produce in a rigorous way an infinite number of general components passing through the point representing the Fermat surface of degree $$\le 9$$ , and many non-reduced components for such degrees.

Unit equations and Fermat surfaces in positive characteristic

In this article we study the three-variable unit equation $x + y + z = 1$ to be solved in $x, y, z \in \mathcal{O}_S^\ast$, where $\mathcal{O}_S^\ast$ is the $S$-unit group of some global function field. We give upper bounds for the height of solutions and the number of solutions. We also apply these techniques to study the Fermat surface $x^N + y^N + z^N = 1$.

Rational curves on a smooth Hermitian surface

We study the set $R$ of nonplanar rational curves of degree $d<q+2$ on a smooth Hermitian surface $X$ of degree $q+1$ defined over an algebraically closed field of characteristic $p>0$, where $q$ is a power of $p$. We prove that $R$ is the empty set when $d<q+1$. In the case where $d=q+1$, we count the number of elements of $R$ by showing that the group of projective automorphisms of $X$ acts transitively on $R$ and by determining the stabilizer subgroup. In the special case where $X$ is the Fermat surface, we present an element of $R$ explicitly.

A Brauer-Siegel theorem for Fermat surfaces over finite fields

We prove an analogue of the Brauer–Siegel theorem for Fermat surfaces over a finite field F q . Namely, letting F d be the Fermat surface of degree d over F q and p g ( F d ) be its geometric genus, we show that, for d → ∞ ranging over the set of integers coprime with q, one has log | Br ( F d ) | · Reg ( F d ) ∼ log q p g ( F d ) ∼ log q 6 · d 3 . Here, Br ( F d ) denotes the Brauer group of F d and Reg ( F d ) the absolute value of a Gram determinant of the Néron–Severi group NS ( F d ) with respect to the intersection form.

Explicit points on y2 + xy − tdy = x3 and related character sums

Let Fq denote a finite field of characteristic p≥5 and let d=q+1. Let Ed denote the elliptic curve over the function field Fq2(t) defined by the equation y2+xy−tdy=x3. Its rank is q when q≡1mod3 and its rank is q−2 when q≡2mod3. We describe an explicit method for producing points on this elliptic curve. In case q≢11mod12, our method produces points which generate a full-rank subgroup. Our strategy for producing rational points on Ed makes use of a dominant map from the degree d Fermat surface over Fq2 to the elliptic surface associated to Ed. We in turn study lines on the Fermat surface Fd using certain multiplicative character sums which are interesting in their own right. In particular, in the q≡7mod12 case, a character sum argument shows that we can generate a full-rank subgroup using μd-translates of a single rational point.

Lines generate the Picard groups of certain Fermat surfaces

We answer a question of T. Shioda and show that, for any positive integer m prime to 6, the Picard group of the Fermat surface Φm is generated by the classes of lines contained in Φm. A few other classes of surfaces are also considered.

RATIONAL POINTS ON SOME FERMAT CURVES AND SURFACES OVER FINITE FIELDS

We give an explicit description of the 𝔽qi-rational points on the Fermat curve uq-1 + vq-1 + wq-1 = 0, for i ∈{1, 2, 3}. As a consequence, we observe that for any such point (u, v, w), the product uvw is a cube in 𝔽qi. We also describe the 𝔽q2-rational points on the Fermat surface uq-1 + vq-1 + wq-1 + xq-1 = 0, and show that the product of the coordinates of any such point is a square.

The transcendental lattice of the sextic Fermat surface

We prove that the integral polarized Hodge structure on the transcendental lattice of a sextic Fermat surface is decomposable. This disproves a conjecture of Kulikov related to a Hodge theoretic approach to proving the irrationality of the very general cubic fourfold.

On the Global Monodromy of a Fibration of the Fermat Surface of Odd Degree $n$

The purpose of this paper is to investigate the global topological monodromy of a certain fibration of the Fermat surface without using numerical analysis by computer.

On the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4

A complete description of the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4 is given. As a by-product we get a positive relation among right hand Dehn twists in the mapping class group of a closed orientable surface of genus 3.

Liquid scintillation detectors for high energy neutrinos

Large liquid scintillation detectors have been generally used for low energy neutrino measurements, in the MeV energy region. We describe the potential employment of large detectors (>1 kiloton) for studies of higher energy neutrino interactions, such as cosmic rays and long baseline experiments. When considering the physics potential of new large instruments the possibility of doing useful measurements with higher energy neutrino interactions has been overlooked. Here we take into account Fermat’s principle, which states that the first light to reach each PMT will follow the shortest path between that PMT and the point of origin. We describe the geometry of this process, and the resulting wavefront, which we call the “Fermat surface”, and discuss methods of using this surface to extract directional track information and particle identification. This capability may be demonstrated in the new long baseline neutrino beam from Jaeri accelerator to the KamLAND detector in Japan. Other exciting applications include the use of Hanohano as a movable long baseline detector in this same beam, and LENA in Europe for future long baseline neutrino beams from CERN. Also, this methodology opens up the question as to whether a large liquid scintillator detector should be given consideration for use in a future long baseline experiment from Fermilab to the DUSEL underground laboratory at Homestake.

A remark on parameterizing nonsingular cubic surfaces

Extending a geometric construction due to Sederberg and to Bajaj, Holt, and Netravali, an algorithm is presented for parameterizing a nonsingular cubic surface by polynomials of degree three. The fact that such a parametrization exists is classical. The present algorithm is, by its purely geometric nature, a very natural one. Moreover, it contains a practical way of finding all lines in an implicitly given cubic surface. Two explicit examples are presented, namely the classical Clebsch diagonal surface and the cubic Fermat surface.

A tessellation for Fermat surfaces in [formula omitted

For each positive integer n , we present a tessellation of C P 2 that can be lifted, through the branched covering, to a symmetric tessellation of the Fermat surface (a 4-manifold) of degree n in C P 3 . The process is systematic and symbolically algebraic. Each four-cell in the tessellation is bounded by four pentahedrons, and each pentahedron has four triangular faces and one quadrilateral face. Graphically, one can produce the entire surface from one single four-cell using translations generated by permutations and phase multiplications of the homogeneous coordinates of C P 3 . Note that the tessellation of the Fermat surface of degree 4, a K3 surface, has exactly 24 vertices.

New strings for old Veneziano amplitudes: III. Symplectic treatment

A d-dimensional rational polytope P is a polytope whose vertices are located at the nodes of Z d lattice. Consider the number | k P ∩ Z d | of points inside the inflated P with coefficient of inflation k ( k = 1 , 2 , 3 , … ) . The Ehrhart polynomial of P counts the number of such lattice points inside the inflated P and (may be) at its faces (including vertices). In Part I [A.L. Kholodenko, New string for old Veneziano amplitudes. I. Analytical treatment, J. Geom. Phys. 55 (2005) 50–74] of our four parts work we noticed that Veneziano amplitude is just the Laplace transform of the generating function (considered as a partition function in the sense of statistical mechanics) for the Ehrhart polynomial for the regular inflated simplex obtained as deformation retract of the Fermat (hyper) surface living in the complex projective space. This observation is sufficient for development of new symplectic (this work) and supersymmetric (Part II) physical models reproducing the Veneziano (and Veneziano-like) amplitudes. General ideas (e.g. those related to the properties of Ehrhart polynomials) are illustrated by simple practical examples (e.g. use of mirror symmetry for explanation of available experimental data on π π scattering, etc.) worked out in some detail. Obtained final results are in formal accord with those earlier obtained by Vergne [M. Vergne, Convex polytopes and quanization of symplectic manifolds, Proc. Natl. Acad. Sci. 93 (1996) 14238–14242].

NEW STRING AMPLITUDES FROM OLD FERMAT (HYPER)SURFACES

The bosonic string theory evolved as an attempt to find a multidimensional analogue of Euler's beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This paper aims to bridge the gap between the mathematical and physical treatments of such beta functions thus providing new topological, algebro-geometric, number-theoretic and combinatorial treatments of the multiparticle Veneziano and Veneziano-like amplitudes. As a result, the entirely new physical meaning of these amplitudes is emerging: they are periods connected with differential forms living on Fermat (hyper)surfaces. Such surfaces are considered as complex projective varieties of Hodge type. Obtained results allow to interpret the particle mass spectrum in terms of the Hodge spectrum of underlying Fermat (hyper)surface. The computational formalism although resembles that used in mirror symmetry calculations employs many additional results from topology, number theory, theory of singularities, etc. It allows to obtain the correlation functions in particle physics and conformal field theories using the same type of the Picard–Fuchs equations.

On the Brauer group

For an arithmetic model of a Fermat surface or a hyperkahler variety with Betti number over a purely imaginary number field , we prove the finiteness of the -components of for all primes . This yields a variant of a conjecture of M. Artin.If is a smooth projective irregular surface over a number field and , then the -primary component of is an infinite group for every prime . Let be the universal family of elliptic curves with a Jacobian structure of level over a number field . Assume that . If is a smooth projective compactification of the surface , then the -primary component of is a finite group for each sufficiently large prime .

On the topological structure of the Fermat surface of degree $5$

On the topological structure of the Fermat surface of degree $5$