Arithmetic mirror symmetry for the 2-torus

Abstract

This paper explores a refinement of homological mirror symmetry which relates exact symplectic topology to arithmetic algebraic geometry. We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the formal disc Spec Z [[q]]. It specializes to a derived equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the curve y 2 + xy = x 3 over Spec Z, the central fibre of the Tate curve; and, over the 'punctured disc' Spec Z ((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2-torus. We also prove that the wrapped Fukaya category of the punctured torus is derived-equivalent over Z to coherent sheaves on the central fiber of the Tate curve.1 Beware: the circumstances under which one expects to find such an X are more subtle than those claimed by our one-sentence précis of Kontsevich's conjecture.We shall need to say what we mean by a CY structure for a category over a commutative ring L. The categories in question are of form H 0 C, where C is an A ∞ -category, and this permits us to make an expedient (but not fully satisfactory) definition: Definition 2.1 A CY structure consists on the L-linear A ∞ -category C consists of cochain-level maps φ A,B : hom C (A, B) ≃ hom C (B, A[n]) ∨ such that the induced maps on cohomology [φ A,B ⊗ 1 F ] : Hom H 0 (C× L F) (A, B) ≃ Hom H 0 (C× L F) (B, A[n]) ∨

A symplectic look at the Fargues-Fontaine curve

“…The more complicated exponent of t in (2.8.8) is the lattice area §1.6 of the same triangle, which coincides with the cardinality of u −1 (D) when D is in the first quadrant very close to (0, 0) -see [LPe2].…”

“…If D is in the first quadrant and extremely close to (0, 0), then this cardinality is given by a simple formula which is independent of D unless the triangle u is extremely acute -let us write area Z (u) for this discretized notion of area. With some additional care, by letting D → (0, 0) (see [LPe2,§7.2.3] and [LPe2, Prop. 9.1]), we get a graded ring…”

“…If L and L ′ are not homologous to zero, one may lift the Maslov index to a Z-valued invariant by equipping L and L ′ (and T ) with gradings, see [LPe2,§6] -let us denote this Z-valued Maslov index by mas Z (x). A formula for the dimension near u ∈ M(x 0 , .…”

A modular compactification of ℳ1,n from A ∞-structures

“…, p n are distinct points of C . In the case n = g , the genus of C , the resulting A ∞ -algebras were studied in [15], [16] (for g = 1) and [21] (in general). The case of genus 0 curves was also studied in [21].…”

“…The above program was implemented in [15] and [21] for the case n = g . In this paper we study a similar equivalence between moduli of curves and A ∞ -structures in the case of curves of arithmetic genus one with n > 1 (smooth) marked points.…”