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The Langlands conjecture for \(\mathrm{GL}_2\) of a local field. (English) Zbl 0469.22013
Let \({\mathbf K}\) be a field and \(k\) be a positive integer. The set of elements in \({\mathbf K}\) that can be written as sums of \(k\)th powers (resp. as sums of \(k\)th powers of totally positive elements) shall be denoted by \(P({\mathbf K},k)\) (resp. \(P^+({\mathbf K},k)\)). \(w({\mathbf K},k)\) (resp. \(g({\mathbf K},k)\)) then denotes the smallest positive integer \(n\) such that each element in \(P({\mathbf K},k)\) (resp. \(P^+({\mathbf K},k)\)) can be written as a sum of \(n\) \(k\)th powers (resp. as a sum of \(n\) \(k\)th powers of totally positive elements), provided such an integer exists. Otherwise, one puts \(w({\mathbf K},k)=\infty\) (resp. \(g({\mathbf K},k)=\infty)\). Recall that by Artin-Schreier theory, the totally positive elements are exactly the nonzero elements in \(P({\mathbf K},2)\). In the literature, \(w({\mathbf K},2)\) is often called the Pythagoras number of \({\mathbf K}\). This paper contains many nice results regarding the invariants \(w\) and \(g\), proved by elementary methods. The paper is mainly based on the author’s PhD thesis from 1970, and having the results available now in such published form is certainly highly appreciated by all those working on these and related problems. Here are some of the main results. One always has \(P^+({\mathbf K},k)=P({\mathbf K},2k)\), and for any odd \(k\) or any non formally real \({\mathbf K}\) one has \({\mathbf K}=P({\mathbf K},k)\) and \(w({\mathbf K},k)<\infty\). If \({\mathbf K}\) is not formally real of characteristic \(0\), then furthermore \(w({\mathbf K},k)=g({\mathbf K},k)<\infty\) for all positive \(k\).
Let \(\alpha\in P({\mathbf K},k)\) and define \(\ell_k(\alpha)\) (resp. \(\ell_k^+(\alpha)\)) to be the smallest integer \(n\) such that \(\alpha\) is a sum of \(n\) \(k\)th powers (resp. of \(n\) \(k\)th powers of totally positive elements). If \({\mathbf K}\) is formally real and \(\alpha\in P({\mathbf K},2)\), the author shows that there exists a positive integer \(\gamma\) only depending on \(\ell_2(\alpha)\) and \(k\), such that \(\ell_k(\alpha)\leq \gamma\) resp. \(\ell_k^+(\alpha)\leq \gamma\) provided certain technical conditions on \(\alpha\) are satisfied. This suffices to show that if \({\mathbf K}\) is an algebraic number field, then \(w({\mathbf K},k)\leq g({\mathbf K},k)<\infty\) for all positive \(k\). Some remarks regarding bounds on \(g(\mathbb{Q},k)\) are added.
The author then turns his attention to the case where the base field is \({\mathbf K}(X)\), the rational function field in one variable over a field \({\mathbf K}\) that has a unique ordering and this ordering is Archimedean. Let \(\partial(f)\) denote the degree of \(f\in{\mathbf K}(X)\). Then it is shown that if \(k>2\) and if \(f\) is positive definite, then \(\ell_k^+(f)\leq g({\mathbf K}(X),k)\) if and only if \(2k\) divides \(\partial(f)\). As consequence, one has for example that for any positive \(k\) and any positive definite polynomial \(f\in {\mathbf K}[X]\) with degree divisible by \(k\), \(f\) is a sum of \(\leq w({\mathbf K}(X),k)\) \(k\)th powers in \({\mathbf K}(X)\). Also, it is shown that if \(w({\mathbf K},2)\) is finite, then \(w({\mathbf K}(X),2)=g({\mathbf K}(X),2)\). The paper concludes with a few examples.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
11F80 Galois representations
11S37 Langlands-Weil conjectures, nonabelian class field theory
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