Brauer points on Fermat curves (Q2730740)

Given a variety \(X(k)\) over a number field \(k\), \textit{Yu. I. Manin} [Actes Congr. Int. Math. (Nice, 1970), Tome 1, 401-411 (1971; Zbl 0239.14010)] showed that elements of the Brauer group \(\text{Br}(X)\) force the \(k\)-rational points of \(X\) to lie in the set of ``Brauer points'' \(X({\mathbb A}_k)^{\text{Br}}\), which is a subset of the set \(X({\mathbb A}_k)\) of all adelic points on \(X\). Manin also proved that in the case where \(X\) is a (smooth, projective, geometrically integral) curve of genus 1 and the Jacobian of \(X\) has finite Shafarevich-Tate group, one has \(X(k) = \emptyset\) if and only if \(X({\mathbb A}_k)^{\text{Br}}=\emptyset\). It follows from a recent preprint of \textit{V. Scharaschkin} [``The Brauer-Manin obstruction for curves'', to appear in Acta Arith.] that if \(X\) is a curve of positive genus having a \(k\)-rational point and if the Mordell-Weil and Shafarevich-Tate groups of the Jacobian of \(X\) are finite, then \(X(k) = X({\mathbb A}_k)^{\text{Br}}\). Beyond this, very little is known about Brauer points on curves. It is conceivable that for every curve \(X\) over a number field \(k\), the closure of \(X(k)\) in \(X({\mathbb A}_k)\) equals \(X({\mathbb A}_k)^{\text{Br}}\). See the discussion on pages 127, 128, and 133 in [\textit{A. Skorobogatov}, Torsors and rational points, Cambridge Univ.\ Press (2001; Zbl 0972.14015)] for further discussion. NEWLINENEWLINENEWLINEThe author proves a result for the Fermat curve \(X^p+Y^p=Z^p\) in \({\mathbb P}^2\) over \(\mathbb Q\), using a modified notion of Brauer points involving only \(F({\mathbb Q}_p)\) instead of the adelic points \(F({\mathbb A}_{\mathbb Q})\). Then it is only the subgroup of elements of \(\text{Br}(F)\) having ``good reduction'' outside \(p\) that gives restrictions on rational points. The author's main result (Theorem A) is that the subset of \(F({\mathbb Q}_p)\) cut out by the restrictions from this subgroup is finite and contains at most \(p\) points, provided that \(\text{rank}_{{\mathbb Z}/p{\mathbb Z}} (C/pC) < (p+3)/8\), where \(C\) is the class group of the cyclotomic field \({\mathbb Q}(e^{2\pi i/p})\). As the author explains, it is likely that the class group bound is always satisfied. NEWLINENEWLINENEWLINEThe main result is deduced from a similar result for quotients of the Fermat curve. The proof uses results from the author's previous three papers on the Fermat curve [Invent. Math. 93, 637-666 (1988; Zbl 0661.14033); Math. Ann. 294, 503-511 (1992; Zbl 0766.14013); Math. Ann. 299, 565-596 (1994; Zbl 0824.14017)].NEWLINENEWLINENEWLINEThe techniques used here (as in some of his previous papers) include Galois cohomology, rigid analysis, and Coleman integrals.