Non-existence of certain Galois representations with a uniform tame inertia weight : A resume (Algebraic Number Theory and Related Topics 2009)
Yoshiyasu Ozeki · Institutional Repositories DataBase (IRDB) · 2011
In this paper, we announce some results on the non-existence of certain semistable Galois representations.We apply them to a conjecture of Rasmussen and Tamagawa. §1. Main resultsOur main concern in this paper is the non-existence of certain semistable Galois representations of a number field.Let \el l be a prime number and K a number field of degree d and discriminant d_{K} .Choose an algebraic closure \ov e r l i ne { K} of K .Fix non- negative integers n, r and w , and a prime number \ell_{0} eq\ell .Put \bul l et :=(n, \ell_{0}, r, w) .Let \ ma t h r m{ R} \ ma t h r m{ e } \ ma t h r m{ p } _ { \ ma t h b b { Q} _ { l } } ( G_ { K} ) be the set of isomorphism classes of n-dimensional \el l-adic representations V of the absolute Galois group G_{ K} =\mat hrm{ G} \mat hrm{ a} 1(\overl i ne{ K} /K) of K which satisfy the following four conditions:(A) For any place $ \ l a mb d a $ of K above \el l , the restriction of V to the decomposition group of (an extension to \ov e r l i ne { K} of) $ \ l a mb d a $ is semistable and has Hodge-Tate weights in [0, r].(B) For some place $ \ l a mb d a $ _ { 0 } of K above \el l _{ 0} , the representation V is unramified at $ \ l a mb d a $ _ { 0 } and the characteristic polynomial \det ( T-\mat hrm{ F} \mat hrm{ r} _{ $\l ambda$_{ 0} } | V) has rational integer coefficients.Furthermore, the roots of the above characteristic polynomial have complex absolute value q _ { $ \ l a mb d a $ _ { 0 } } w/2 for every embedding \ o v e r l i n e { \ m a t h b b { Q } } _ { \ e l l } into \ ma t h b b { C} .Here \ m a t h r m { F } \ m a t h r m { r } _ { $ \ l a m b d a $ _ { 0 } } and q_ { $\l ambda$_ { 0} } are the arithmetic Frobenius and the order of the residue field of $ \ l a mb d a $ _ { 0 } , respectively.