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On the integrality of some Galois representations
| Content Provider | Semantic Scholar |
|---|---|
| Author | Gross, Robert |
| Copyright Year | 1995 |
| Abstract | We find an appropriate topology to put on K, the fraction field of the Iwasawa algebra Λ = Zp[[T ]], so that compact subgroups of K are in fact contained in Λ. This ensures that Galois representations to K have image in Λ. Let Λ = Zp[[T ]] be the Iwasawa algebra. Λ is a unique factorization domain. The p-adic Weierstrass Preparation Theorem says that elements of Λ may be represented as uf , where f is a polynomial and u is a unit. Let M = (p, T ) be the maximal ideal of Λ. Topologize Λ so that a base of neighborhoods of 0 is given by powers of M , and define neighborhoods of other elements of Λ by translation. Let K be the field of fractions of Λ. The first question to consider is how to topologize K. One somewhat obvious approach is to say that a set U ⊆ K is open in K precisely when kU ∩Λ is an open subset of Λ for all k ∈ K. This definition makes addition and multiplication continuous. Topologized in this way, a compact subset of GLn(K) which is also a subgroup is conjugate to a subset of GLn(Λ). Unfortunately, there is one major drawback to this topology. Proposition. The function f(x) = x is not continuous in this topology. Proof. There are many ways to see this. Perhaps the simplest is to observe that the sequence an = p+ T n converges to p. However, the sequence a n is closed, since for a fixed k ∈ K , ka n will be an element of Λ for only finitely many n. Hence, a n cannot converge to p . We therefore need a different topology on K, and fortunately there is an obvious candidate. If λ ∈ Λ, we can define v(λ) = n if λ ∈ M and λ 6∈ M and v(0) = ∞. Krull’s Theorem [1] implies that ⋂ M = {0}, and so the function v is well-defined. Lemma. v is a valuation on Λ. Proof. Let f, g ∈ Λ. Set v(f) = m and v(g) = n. Obviously, v(f + g) ≥ min(v(f), v(g)), so we need only show that v(fg) = v(f) + v(g). Use the Weierstrass Preparation Theorem to write f = u1f , g = u2g , where u1 and u2 are units and f ′ and g are polynomials. Write f ′ = ∑ aiT i and g = ∑ bjT . Let vp be the usual p-adic valuation on Zp. Of those terms in ∑ aiT i with v(aiT ) = m, let akT k be the term so that vp(ak) is minimal. (It is easy to see that there is a unique minimum, because if v(aiT ) = m, then vp(ai) = m− i.) Similarly, let blT l be the term in the second sum minimizing vp(bl) subject to v(blT ) = n. If we now consider the coefficient ck+l of T k+l in the product fg = u1u2f g, we see that vp(ck+l) = vp(ak) + vp(bl). Therefore v(ck+lT ) = m+ n, and we finally have v(fg) = m+ n. This lemma in fact is true in considerably greater generality, but the statement does not seem to appear in the literature in this form. 1991 Mathematics Subject Classification. 22C05, 11S20. |
| Starting Page | 299 |
| Ending Page | 301 |
| Page Count | 3 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1995-1215201-0 |
| Alternate Webpage(s) | http://fmwww.bc.edu/gross/galois.pdf |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1995-123-01/S0002-9939-1995-1215201-0/S0002-9939-1995-1215201-0.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1995-1215201-0 |
| Volume Number | 123 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
