SPECIAL VALUES OF THE ZETA FUNCTION OF AN ARITHMETIC SURFACE

We prove that the special-value conjecture for the zeta function of a proper, regular, flat arithmetic surface formulated in [6] at $s=1$ is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of the generic fibre. There are two key results in the proof. The first is the triviality of the correction factor of [6, Conjecture 5.12], which we show for arbitrary regular proper arithmetic schemes. In the proof we need to develop some results for the eh-topology on schemes over finite fields which might be of independent interest. The second result is a different proof of a formula due to Geisser, relating the cardinalities of the Brauer and the Tate–Shafarevich group, which applies to arbitrary rather than only totally imaginary base fields.

Grothendieck-Serre duality and theta-invariants on arithmetic surfaces

In the paper, a description of the Grothendieck-Serre duality on an arithmetic surface by means of fixing a horizontal divisor is given and this description is applied to the generalization of theta-invariants.

Constructing elliptic curves from Galois representations

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius elements, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.

Effects of Process Parameters on Material Removal Rate and Arithmetic Surface Roughness in Rubber Carbon Black Cutting by EDM

This research proposes a new application of Electrical Discharge Machine (EDM) for cutting the rubber carbon black. The new material of this study was a natural rubber mixed with carbon black for increased electrical conductivity of natural rubber. was accomplished through the technique of design and analysis of experiment and statistical analysis. The 24 Full factorial design was selected to conduct the experiment to determine the optimal mixed rubber cutting process parameters. The study has been carried out on the influence of design factors of the concentrate amount of carbon black per hundred of natural rubber (PHR), the voltage across the gap between and electrode tool and workpieces (Volt), the value of the discharge current (Ampere), and the duration of time (On-time, μs) that current is allowed to flow per cycle over response variables arithmetic surface roughness and material removal rate. The experimental results have shown the value of the discharge current that significantly affect to the degree of material removal rate of rubber mixed with carbon black and the duration of time that significantly affect to the degree of arithmetic surface roughness.

Linear projections and successive minima

LetXbe an arithmetic surface, and letLbe a line bundle onX. Choose a metrichon the lattice Λ of sections ofLoverX. When the degree of the generic fiber ofXis large enough, we get lower bounds for the successive minima of (Λ,h) in terms of the normalized height ofX. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison’s proof that smooth projective curves of high degree are Chow semistable.

Linear projections and successive minima

Let X be an arithmetic surface, and let L be a line bundle on X. Choose a metric h on the lattice Λ of sections of L over X. When the degree of the generic fiber of X is large enough, we get lower bounds for the successive minima of (Λ,h) in terms of the normalized height of X. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison’s proof that smooth projective curves of high degree are Chow semistable.