The field with one element

Fifteen lessons on the field with one element. Weil’s proof for curves and what is missing over the integers proves the Riemann hypothesis for curves over a finite field by intersection numbers on the square of the curve, including a proof of the Hodge index theorem, and describes the objects that a proof of the same kind would need over the integers. The next lessons study algebraic structures that have been proposed as a base below the integers. Monoids with zero keep the multiplication of a ring and forget its addition. Glued like rings, they give monoid schemes; over an algebraically closed field their base changes are exactly the toric varieties. Toën and Vaquié define schemes relative to a symmetric monoidal category, and for sets they recover monoid schemes. Borger’s Λ-rings, rings with commuting Frobenius lifts, give another way to descend from the integers. Durov’s generalized rings are finitary commutative monads on sets; they include the unit ball of the real numbers and give a compactification of \(\operatorname{Spec}\mathbb Z\). Lorscheid’s blueprints keep a monoid together with a relation that records sums, and they give a Tits–Weyl model of \(\mathrm{SL}_2\). In characteristic one, where \(1+1=1\), the ordinary addition of positive numbers is recovered from the maximum by the entropy formula, and Connes’s Witt construction turns a perfect semiring of characteristic one, with a parameter, into an algebra over the real numbers; hyperrings arise as quotients of rings by groups of units, among them the adèle class space. Γ-sets keep a record of which families can be added; over the sphere \(\mathbb S\) they lead to the Riemann–Roch theorem of Connes and Consani for the compactification of \(\operatorname{Spec}\mathbb Z\). The arithmetic site of Connes and Consani is the topos of sets with an action of the positive integers, with the tropical integers as structure sheaf; its points over \(\mathbb R_{\max}\) form the quotient \(\mathbb Q^\times\backslash\mathbb A_{\mathbb Q}/\widehat{\mathbb Z}^\times\) of the adèle class space, and its square carries Frobenius correspondences. The projective line over \(\mathbb F_1\) and the ABC conjecture proves the Riemann–Hurwitz formula and the theorem of Mason and Stothers, constructs Smirnov’s projective line over \(\mathbb F_1\) and the maps \(\tau_q\), proves that Smirnov’s Hurwitz inequality implies the ABC conjecture, and describes these objects in monoid geometry following Jarra. Each lesson has complete proofs, examples, and exercises with solutions.

Prerequisites: commutative rings and their prime spectra, sheaves on a topological space, and categories and functors; for Weil’s proof also algebraic curves, coherent cohomology and the Riemann zeta function; for monoid schemes also varieties over a field, normality and divisors; for the Witt construction metric spaces and compactness. Facts from these subjects are cited by tag from the Stacks project, for real analysis from the open text of the core course Real Analysis I, and from the lessons on the Riemann zeta function and on the Stone–Weierstrass theorem of this collection.

Reading order

The reading order includes counting over finite fields, torified varieties, varieties after Soulé and Connes–Consani, and the scaling site. All fifteen lessons are included in the download with their editable sources. Linked proofs in other courses and external references are not bundled.

Download the lessons and their editable sources · Sources and authorship · Course record

Written by Claude Opus 5.5 (Anthropic) and self-checked by the writing AI; the line below each lesson title states its further checks. Public domain (CC0 1.0).