On the Mumford-Tate conjecture for Abelian varieties with reduction conditions

In this thesis we study Galois representations corresponding to abelian varieties with certain reduction conditions. We show that these conditions force the image of the representations to be "big," so that the Mumford-Tate conjecture (:= MT) holds. We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding moduli space.

The main results of the thesis are the following two theorems.

Theorem A: Let A be an absolutely simple abelian variety, End° (A) = k : imaginary quadratic field, g = dim(A). Assume either dim(A) ≤ 4, or A has bad reduction at some prime ϕ, with the dimension of the toric part of the reduction equal to 2r, and gcd(r,g) = 1, and (r,g) ≠ (15,56) or (m -1, m(m+1)/2). Then MT holds.

Theorem B: Let M be the moduli space of abelian varieties with fixed polarization, level structure and a k-action. It is defined over a number field F. The subset of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable reduction at a large enough prime ϕ of F is dense in M(C) in the complex topology. In particular, the set of simple abelian varieties having bad reductions with fixed dimension of the toric parts is dense.

Besides this we also established the following results:

(1) MT holds for some other classes of abelian varieties with similar reduction conditions. For example, if A is an abelian variety with End° (A) = Q and the dimension of the toric part of its reduction is prime to dim( A), then MT holds.

(2) MT holds for Ribet-type abelian varieties.

(3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds.

(4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some 4-folds of type I.

(5) For some abelian varieties either MT or the Hodge conjecture holds.

Electron energy-loss spectroscopy study of polyatomic molecular systems under pyrolytic conditions

The technique of variable-angle, electron energy-loss spectroscopy has been used to study the electronic spectroscopy of the diketene molecule. The experiment was performed using incident electron beam energies of 25 eV and 50 eV, and at scattering angles between 10° and 90°. The energy-loss region from 2 eV to 11 eV was examined. One spin-forbidden transition has been observed at 4.36 eV and three others that are spin-allowed have been located at 5.89 eV, 6.88 eV and 7.84 eV. Based on the intensity variation of these transitions with impact energy and scattering angle, and through analogy with simpler molecules, the first three transitions are tentatively assigned to an n → π* transition, a π - σ* (3s) Rydberg transition and a π → π* transition.

Thermal decomposition of chlorodifluoromethane, chloroform, dichloromethane and chloromethane under flash-vacuum pyrolysis conditions (900-1100°C) was investigated by the technique of electron energy-loss spectroscopy, using the impact energy of 50 eV and a scattering angle of 10°. The pyrolytic reaction follows a hydrogen-chloride α-elimination pathway. The difluoromethylene radical was produced from chlorodifluoromethane pyrolysis at 900°C and identified by its X^1 A_1 → A^1B_1 band at 5.04 eV.

Finally, a number of exploratory studies have been performed. The thermal decomposition of diketene was studied under flash vacuum pressures (1-10 mTorr) and temperatures ranging from 500°C to 1000°C. The complete decomposition of the diketene molecule into two ketene molecules was achieved at 900°C. The pyrolysis of trifluoromethyl iodide molecule at 1000°C produced an electron energy-loss spectrum with several iodine-atom, sharp peaks and only a small shoulder at 8.37 eV as a possible trifluoromethyl radical feature. The electron energy-loss spectrum of trichlorobromomethane at 900°C mainly showed features from bromine atom, chlorine molecule and tetrachloroethylene. Hexachloroacetone decomposed partially at 900°C, but showed well-defined features from chlorine, carbon monoxide and tetrachloroethylene molecules. Bromodichloromethane molecule was investigated at 1000°C and produced a congested, electron energy-loss spectrum with bromine-atom, hydrogen-bromide, hydrogen-chloride and tetrachloroethylene features.

I. Foehn winds of southern California. II. Foehn wind cyclo-genesis. III. Weather conditions associated with the Akron disaster. IV. The Los Angeles storm of December 30, 1933 to January 1, 1934

I. Foehn winds of southern California.
An investigation of the hot, dry and dust laden winds occurring in the late fall and early winter in the Los Angeles Basin and attributed in the past to the influences of the desert regions to the north revealed that these currents were of a foehn nature. Their properties were found to be entirely due to dynamical heating produced in the descent from the high level areas in the interior to the lower Los Angeles Basin. Any dust associated with the phenomenon was found to be acquired from the Los Angeles area rather than transported from the desert. It was found that the frequency of occurrence of a mild type foehn of this nature during this season was sufficient to warrant its classification as a winter monsoon. This results from the topography of the Los Angeles region which allows an easy entrance to the air from the interior by virtue of the low level mountain passes north of the area. This monsoon provides the mild winter climate of southern California since temperatures associated with the foehn currents are far higher than those experienced when maritime air from the adjacent Pacific Ocean occupies the region.

II. Foehn wind cyclo-genesis.
Intense anticyclones frequently build up over the high level regions of the Great Basin and Columbia Plateau which lie between the Sierra Nevada and Cascade Mountains to the west and the Rocky Mountains to the east. The outflow from these anticyclones produce extensive foehns east of the Rockies in the comparatively low level areas of the middle west and the Canadian provinces of Alberta and Saskatchewan. Normally at this season of the year very cold polar continental air masses are present over this territory and with the occurrence of these foehns marked discontinuity surfaces arise between the warm foehn current, which is obliged to slide over a colder mass, and the Pc air to the east. Cyclones are easily produced from this phenomenon and take the form of unstable waves which propagate along the discontinuity surface between the two dissimilar masses. A continual series of such cyclones was found to occur as long as the Great Basin anticyclone is maintained with undiminished intensity.

III. Weather conditions associated with the Akron disaster.
This situation illustrates the speedy development and propagation of young disturbances in the eastern United States during the spring of the year under the influence of the conditionally unstable tropical maritime air masses which characterise the region. It also furnishes an excellent example of the superiority of air mass and frontal methods of weather prediction for aircraft operation over the older methods based upon pressure distribution.

IV. The Los Angeles storm of December 30, 1933 to January 1, 1934.
This discussion points out some of the fundamental interactions occurring between air masses of the North Pacific Ocean in connection with Pacific Coast storms and the value of topographic and aerological considerations in predicting them. Estimates of rainfall intensity and duration from analyses of this type may be made and would prove very valuable in the Los Angeles area in connection with flood control problems.