On uniform lower bound of the Galois images associated to elliptic curves

Keisuke Arai

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A local analogue of the Grothendieck Conjecture is an equivalence between the category of complete discrete valuation fields $K$ with finite residue fields of characteristic $p \neq 0$ and the category of absolute Galois groups of fields $K$ together with their ramification filtrations. The case of characteristic 0 fields $K$ was studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case $p > 2$ (but with no restrictions on the characteristic of $K$ )....

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A large number of papers have contributed to determining the structure of the tame kernel $K ₂_{F}$ of algebraic number fields F. Recently, for quadratic number fields F whose discriminants have at most three odd prime divisors, 4-rank formulas for $K ₂_{F}$ have been made very explicit by Qin Hourong in terms of the indefinite quadratic form x² - 2y² (see [7], [8]). We have made a successful effort, for quadratic number fields F = ℚ (√(±p₁p₂)), to characterize in terms of positive definite binary quadratic...