Goro Kato

Zeta matrices of elliptic curves

Abstract Let O = lim n Z/p n Z , let A = O[g 2 , g 3 ] Δ , where g2 and g3 are coefficients of the elliptic curve: Y2 = 4X3 − g2X − g3 over a finite field and Δ = g23 − 27g32 and let B = A [X, Y] (Y 2 − 4X 3 + g 2 X + g 3 ) . Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free A † ⊗ Z Q -module H 1 (X, A † ⊗ Z Q) . Main results are; Theorem 1.1: X2 dY and Y dX are basis elements for H 1 ( X , Γ A ∗ ( X ) † ⊗ Z Q) ; Theorem 1.2: Y dX, X2 dY, Y−1 dX, Y−2 dX and XY−2 dX are basis elements for H 1 ( X − (Y = 0), Γ A ∗ ( X ) † ⊗ Z Q) , where X is a lifting of X, and all the necessary recursive formulas for this explicit computation are given.

Topos Theoretic Approach to Space and Time

At the level of the Planck scale (around \(10^{-33}{\text {cm}} \)) and beyond, i.e., a sub-Planckian domain (less than \(10^{-43}\) s after so called big bang), the usual concept of space and time becomes uncertain where the gravitational field might be simply a quantum fluctuation of a vacuum. Elementary particles or strings, the fundamental entities for our universe, are difficult to be considered at such a microcosm level. An emergence occurs when a physical phenomenon is the consequence of organization from the given local information data. Namely, we cannot tell any difference between electrons in a human brain and in an apple. This is just as we cannot tell any difference between a note in a piece by Mozart and a note in a piece by Bach. We use the concept of a sheaf as the device from local to global transition. In order to formulate space and time for those microcosm domains in terms of sheaves providing a background free notion in the sense of quantum gravity, the notions of the associated (pre)sheaves of time, space, and matter are introduced in the following sense. For a particle \(\overline{m}\), we assign an associated presheaf m with \(\overline{m}\). A presheaf is by definition a contravariant functor from a site (i.e., a category with a Grothendieck topology) to a product category. This is the notion of the temporal topos theory abbreviated as t-topos theory developed in [1, 2, 3, 4, 5]. For space and time, we associate a combined sheaf \(\omega =\left( \kappa ,\tau \right) \) where space sheaf \(\kappa \) and time sheaf \(\tau \) are considered to be t-entangled in the sense that both sheaves behave as one sheaf. With the notions of sheaves and categories, we will give the formulations for the uncertainty principle, particle-wave duality, and t-entanglement together with the relativistic concept of a \(t \) -light cone (or an ur-light cone) valid in macrocosm and microcosm. As a consequence of the topos theoretic formulations, the possible scenario of pre and primitive stages of a universe, i.e., ur-big bangs in terms of t-topos theory will be provided. The main concepts to formulate these notions are coming from categorical notions of a micro-decomposition of a presheaf and a micro-covering of a t-site object.