Citation
Lesin, Alexander Abraham (1994) On the MumfordTate Conjecture for Abelian Varieties with Reduction Conditions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/g2mpjn54. https://resolver.caltech.edu/CaltechTHESIS:05082013153012294
Abstract
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 MumfordTate 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 kaction. 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 Ribettype abelian varieties.
(3) The Hodge and the Tate conjectures are equivalent for abelian 4folds.
(4) MT holds for abelian 4folds of type II, III, IV (Theorem 5.0(2)) and some 4folds of type I.
(5) For some abelian varieties either MT or the Hodge conjecture holds.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Awards:  The W.P. Carey and Co., Inc., Prize in Applied Mathematics, 1994. 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  2 May 1994 
Record Number:  CaltechTHESIS:05082013153012294 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:05082013153012294 
DOI:  10.7907/g2mpjn54 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  7683 
Collection:  CaltechTHESIS 
Deposited By:  INVALID USER 
Deposited On:  08 May 2013 22:50 
Last Modified:  17 Jul 2020 18:26 
Thesis Files

PDF
 Final Version
See Usage Policy. 10MB 
Repository Staff Only: item control page