An Effective Version of the Grunwald-Wang Theorem

Citation

Wang, Song (2002) An Effective Version of the Grunwald-Wang Theorem. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/dmev-ny09. https://resolver.caltech.edu/CaltechTHESIS:02012012-151008104

Abstract

The main purpose of my thesis is to establish an effective version of the Grunwald-Wang Theorem, which asserts that given a family of local characters X_v of K^*_v of exponent m where v ϵ S for a finite set S of primes of K, there exists a global character X of exponent m (unless some special case occurs, when it is 2m) whose component at v is X_v. The effectivity problem for this theorem is to bound the norm N(X) of the conductor of X in terms of K, m, S and N(X_v) . This problem was encountered in 1995 in the work of Hoffstein and Ramakrishnan ([H-Ra95]), where they needed it in a particular case when K ∩ µ_m AND m is a prime. In this thesis, we solve this problem completely, and show in the general case that N(_X) is bounded by A∏_(vϵS) N(X_v)^B with A = (A_0N_s)^(C1|s|). In the special cases when K ∩ µ_m OR m is a prime, we can give a better bound for N(_X) with A = A_0N^(C_2)_S, where A_0, B, C_1 and C_2 are independent of S. The later bound improves the result of [H-Ra95]. We get an even more precise bound, namely N(_X) ≤ 4 ∏_(pϵS) NpN(X_p), when K = Q and m = 2.

In this thesis, we develop three different approaches, dealing with the quadratic extension case, the Kummer or the general l-extension case, and the general case respectively. In addition to class field theory, we use a reduction process to the unramified case and certain modified effective versions of the Chebotarev Density Theorem. Also, in the general case, we transport to this problem some techniques from Algebra involving essential subgroups and essential closures.

To check the maximal range of our method, we also consider the problem with GRH, and get A = (A_0 log N_s)^(C_0|S|) in the general case where A_0 and C_0 are independent of S. When K = Q and m = 2, we do even better with A « (2^(|S|)logN_s)^2. To get these results, we use yet another modification of the effective version of the Chebotarev Density Theorem (with GRH).

These effective results have some interesting applications in concrete situations. To give a simple example, if we fix p and l, one gets a good least upper bound for N such that p is not an l-th power mod N. One also gets the least upper bound for N such that l^τ |ø(N) and p is not an l-th power mod N. The table 3.1 describes the best least upper bound we get this way for quadratic Dirichlet characters on C_Q having desired local behavior at some p and/or infinity.

Thesis Files