Principal bundles and reciprocity laws in number theory

We give a brief survey of some ideas surrounding non-abelian Poitou-Tate duality in the setting of arithmetic moduli schemes of principal bundles for unipotent fundamental groups and their Diophantine applications. 1. Principal bundles and their moduli Moduli spaces of principal bundles (or torsors) have played a prominent role in geometry, topology, and mathematical physics over the last half-century [2, 12, 21, 25]. However, it would appear that arithmetic applications predate these developments by many decades. A prominent example is Weil’s work on the Jacobian JX of an algebraic curve X [23]. While its analytic construction had been known since the 19th century, Weil gave an algebro-geometric construction so that the inclusion X ⊂ JX that sends x to the class of the line bundle OX(x)⊗OX(−b) might be used to study the arithmetic of X. In Weil’s approach, when X is defined over a number field F , so is JX . Furthermore, choosing an F -rational basepoint b ∈ X(F ), rationality is preserved by the inclusion, suggesting the possibiity of studying X(F ) via the superset JX(F ). This research resulted in the Mordell-Weil theorem, stating that JX(F ) is finitely-generated, a result which then was generalised to arbitrary abelian varieties. Weil hoped to prove that the geometric intersection X ∩ JX(F ) is finite, thereby proving the Mordell conjecture. However, the abelian nature of JX(F ), a useful property in itself, turned out to be an obstruction more than a help when applied to the arithmetic of X. Nevertheless, the Jacobian was subsequently used by Siegel to prove the finiteness of integral points on affine curves over number fields, thereby convincing arithmeticians of the utility of this abstract construction. Later, Weil attempted to move beyond the abelian framework by considering moduli spaces Bunn(X) of vector bundles of rank n over X [24]. Serre [20] describes this work in his obituary for Weil as ‘a text presented as analysis, whose significance is essentially algebraic, but whose motivation is arithmetic.’ He correctly stresses the visionary nature of the paper, written long before the advent of geometric invariant theory made it possible to give a systematic treatment of such moduli spaces. Today, they play an important role in various geometric versions of 1991 Mathematics Subject Classification. 14G10, 11G40, 81T45 . Supported by grant EP/M024830/1 from the EPSRC. c ©0000 (copyright holder)