Gross-Schoen cycles and dualising sheaves (Q2655167)

Let \(X\) be a smooth projective geometrically connected curve over a field \(k\) of genus \(g>1\). Let \(Y=X^3\) be the triple product of \(X\) over \(k\). For a divisor \(e\) on \(X\) with rational coefficients of degree one, Gross and Schoen constructed a cycle \(\Delta_e\) in the homologically trivial part \(\mathrm{Ch}^2(Y)^0\) of the Chow group of \(Y\) with rational coefficients. The non-triviality of \(\Delta_e\) is detected by the height \(\langle \Delta_e, \Delta_e \rangle\) of \(\Delta_e\), which is defined if either \(k\) is a number field, or if \(k\) is a function field of a smooth projective irreducible curve \(B\) over a field. The height \(\langle \Delta_e, \Delta_e \rangle\) reaches its minimal value when \(e - \xi\) is a torsion divisor, where \(\xi\) is a divisor on \(X\) such that \((2g-2)\xi\) is the canonical divisor. The main result of this paper expresses \(\langle \Delta_{\xi}, \Delta_{\xi} \rangle\) in terms of the self intersection \(\omega^2\) of the relative dualising sheaf, plus some contribution from local (bad) places of \(k\). This result has a lot of important applications which include (among others) a new approach toward the effective Bogomolov conjecture and the Northcott property for \(\Delta_{\xi}\) and for the Ceresa cycle \(X - [-1]^* X\). Here we have a look at an application to the \(L\)-function: Assume that \(k\) is the function field of \(B\) as above, and that \(X\) can be extended to a non-isotrivial family of smooth projective curves over \(B\). Then, one has \(\langle \Delta_{\xi}, \Delta_{\xi} \rangle > 0\) (because all local contribution vanish in this situation, and because \(\omega^2 >0\)). When \(B\) is a curve of genus \(\geq 3\) over a finite field, this in turn implies that the order of \(L(M, s)\) at \(s=0\) is \(> 0\), where \(M\) is the motive whose \(\ell\)-adic realization is given by the kernel of \(\bigwedge^3 H^1(X)(2) \to H^1(X)(1), ~ a \wedge b \wedge c \mapsto a(b \cup c) + b(c \cup a) + c(a \cup b)\). (Here \(\cup : H^1(X) \otimes H^1(X) \to \mathbb{Q}_l(-1)\) is the Weil pairing.) Actually, this order is shown to be \(\geq 2\), since the sign of the functional equation is \(+1\). In view of Tate's conjecture, there should be another element in \(\mathrm{Ch}^2(M)^0\) which is linearly independent of \(\Delta_{\xi}\). It is left as a (very interesting) question how to find such a cycle.