Mirror congruence for rational points on Calabi-Yau varieties (Q2373593)

This paper discusses problems in arithmetic mirror symmetry of a mirror pair of Calabi--Yau varieties. The main results are concerned with congruences involving number of rational points on a mirror pair of Calabi--Yau varieties. Let \({\mathbb{F}}_q\) denote the finite field of \(q\) elements of characteristic \(p\), and let \(W=W({\mathbb{F}}_q)\) be the ring of Witt vectors of \({\mathbb{F}}_q\). Theorem 1. Let \(X_0\) be a smooth projective variety defined over \({\mathbb{F}}_q\). Suppose that \(X_0\) has a smooth projective lifting \(X\) to \(W\) such that the \(W\)-modules \(H^r(X,\Omega^s_{X/W})\) are free. Let \(G\) be a finite group of \(W\)-automorphisms acting on \(X\) from right. Suppose \(G\) acts trivially on \(H^i(X,{\mathcal{O}}_X)\) for all \(i\). Then for any natural number \(k\), we have the congruence between the number of rational points on \(X_0\) and \(X_0/G\) over \({\mathbb{F}}_{q^k}\): \[ \#X_0({\mathbb{F}}_{q^k})\equiv \#(X_0/G)({\mathbb{F}}_{q^k})\pmod{q^k} \] Now apply Theorem 1 to a mirror pair of Calabi-Yau varieties. Theorem 2. Let \(X_0\) be a geometrically connected Calabi--Yau variety of dimension \(n\) and let \(X\) be its lifting to \(W\) satisfying the condition of Theorem 1. Let \(G\) be a finite group of \(W\)-automorphisms of \(X\). Assume that \(G\) fixes a non-zero \(n\)-form on \(X\). Then for every natural number \(k\), we have the congruence \[ \#X_0(\mathbb{F}_{q^k})\equiv\#(X_0/G)(\mathbb{F}_{q^k})\pmod{q^k}. \] Example. Of particular interest is the Calabi-Yau variety defined by the Dwork family \[ x_0^{n+1}+\cdots+x_n^{n+1}+\lambda \prod_{i=0}^n x_i=0\subset {\mathbb{P}}^n_{{\mathbb{F}}_q}\quad{\text{where}}\quad\lambda\in{\mathbb{F}}_q. \] Let \[ G=\{(\zeta_0,\cdots,\zeta_n)\,|\, \zeta_i\in{\mathbb{F}}_q,\, \zeta_i^{n+1}=1,\,\prod_{i=0^n}\zeta_i=1\}. \] Consider the action \(G\times X_0\to X_0\) defined by \[ (\zeta_0,\cdots,\zeta_n)\times [x_0:\cdots:x_n]\mapsto [\zeta_0x_0:\cdots:\zeta_nx_n]. \] Then \(X_0\) and \(X_0/G\) form a mirror pair, and for any natural number \(k\), we have the congruence \[ \#X_0({\mathbb{F}}_{q^k})\equiv \#(X_0/G)({\mathbb{F}}_{q^k})\pmod{q^k}. \] Now let \(X\) be a smooth projective Calabi-Yau variety and \(Y\) be a smooth crepant resolution of \(X/G\) which is again a Calabi-Yau variety. Then arithmetic mirror symmetry conjecture for a mirror pair \((X,Y)\) of Calabi-Yau varieties may be reduced to establishing the congruence \[ \#(X/G){\mathbb{F}}_{q^k})\equiv \#Y({\mathbb{F}}_{q^k})\pmod{q^k} \] for every natural number \(k\). Further information about the number of rational points is obtained for geometrically connected varieties with the property \(H^i(X,{\mathcal{O}}_X)=0\) for all \(i\neq 0\). Theorem 3. Let \(X_0\) be a smooth geometrically connected projective variety over \({\mathbb{F}}_q\). Suppose that \(H^i(X_0,{\mathcal{O}}_{X_0})=0\) for all \(i\neq 0\). Then for any natural number \(k\), \[ \#X_0({\mathbb{F}}_{q^k})\equiv 1\pmod{q^k}. \] If \(G\) is a finite group of \({\mathbb{F}}_q\)-automorphisms on \(X_0\). Then for any natural number \(k\), \[ \#(X_0/G)({\mathbb{F}}_{q^k})\equiv \#X_0({\mathbb{F}}_{q^k})\pmod{q^k}. \] The liftability condition in Theorem 1 is crucial. However, if the order of \(G\) is not divisible by the characteristic \(p\), this condition may be dropped. The proofs follow from establishing congruences for traces of \(F^k\) and \(gF^k\) on crystalline cohomology groups of \(X_0/W\). Here \(F\) is the Frobenius correspondence and \(g: X_0\to X_0\) is an \({\mathbb{F}}_q\)-automorphism of finite order.