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Algebraic spectral curves over \(\mathbb{Q}\) and their tau-functions. (English) Zbl 1484.14073

Donagi, Ron (ed.) et al., Integrable systems and algebraic geometry. A celebration of Emma Previato’s 65th birthday. Volume 2. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 459, 41-91 (2020).
Consider a matrix polynomial of the form \[ W(z) = B^0 z^m + \cdots + B^m, \quad B^i \in \mathrm{Mat}_n(\mathbb{C}). \] Let \(C \to \mathbb{CP}^1\) be the associated spectral curve whose affine part is given by \[ \{(z,w)\in \mathbb{C}^2 ~|~\det\left(w\cdot \mathbf{1}_n - W(z)\right) = 0\}. \] If \(W\) is generic, then \(C\) is smooth and irreducible with \(n\) pairwise distinct points \(P_1, \dots, P_n\) over \(\infty\). Note that the spectral curve \(C\) comes with a natural line bundle \(\mathcal{L}\) defined by the eigenvectors of \(W\). Finally, we denote by \(\theta\) the Riemann theta function of \(C\) associated with a fixed canonical basis of \(H_1(C,\mathbb{Z})\).
The present article studies \(N\)-differentials \(\nu_{N, \mathbb{Q}}\) on \(C\) for \(N\geq 2\) and \(N\)-tuples \(\mathbb{Q}\) of points on \(C\). The \(\nu_{N, \mathbb{Q}}\) are defined by the \(N\)-th logarithmic differential of \(\theta\) at the point in the Jacobian corresponding to \(\mathcal{L}\) (after tensoring with an appropriate divisor) and the \(N\)-tuples \(\mathbb{Q}\). The surprising main result of this articles expresses each \(\nu_{N, \mathbb{Q}}\) explicitly in terms of \(W\) and \(\mathbb{Q}\) only. If \(W\) has rational coefficients, i.e. \(B^i \in \mathrm{Mat}_n(\mathbb{Q})\), then it is further shown as a corollary that \(\nu_{N, \mathbb{Q}}\) has only rational coefficients when expanded around \(\infty\).
The key of proving these statements (for \(N \geq 3\); for \(N=2\) the methods are related but more direct) is the relationship to the \(n\)-wave (or AKNS-D) hierarchy defined by \[ [L_{a, k}, L_{b,l}] = 0,\quad L_{a, k} = \frac{\partial}{\partial t^a_k} - U_{a, k}(\mathbf{t}; z). \] Here \(U_{a, k}(\mathbf{t}; z)\) are any \(n \times n\)-matrix-valued polynomials in \(z\) of degree \(k+1\) of a special form. Now given a spectral curve \(C\) as above and \(N\geq 0\), a solution to the \(n\)-wave hierarchy is constructed (Proposition 2.20) following Krichever’s approach and using a vector-valued Baker-Akhiezer function. We note that the \(U_{a,k}\) of the hierarchy are in fact constructed from the spectral curve \(C\). A key result (Proposition 2.24) is that the tau-function attached to such a solution (reviewed in Appendix A) can be expressed as \[ \tau(\mathbf{t}) = a(\mathbf{t})\cdot \theta(V(\mathbf{t}) - \mathbf{u}_0). \] Here \(\theta\) is the Riemann theta function of \(C\) as above and \(a(\mathbf{t})\), \(V(\mathbf{t})\) are certain scalar-/vector-valued functions.
From the previous formula it follows that the \(N\)-differentials \(\nu_{N, \mathbb{Q}}\) are equivalently defined as certain \(N\)-th logarithmic differentials of the tau-function \(\tau(\mathbf{t})\). Applying results on such differentials for more general solutions of the \(n\)-wave hierarchy (proven in Appendix A), it is possible to express them in terms of \(W(z)\) for the solutions attached to \(C\). Thereby the main theorem follows (see end of Section 2.3).
Despite being technical, this article is well written and organized. It contains several interesting results along the way. For example, results on the divisor of normalized eigenvectors of \(W\) (see Section 2) and its relation to the relative Jacobian \(J(C; P_1, \dots, P_n)\) (which can be considered as the Jacobi variety of the singular curve obtained from \(C\) by identifying the points \(P_i\) over \(\infty\)). Finally, explicit examples of the main results are provided as well as an appendix on tau-functions of solutions to the \(n\)-wave hierarchy.
For the entire collection see [Zbl 1456.14004].

MSC:

14H70 Relationships between algebraic curves and integrable systems
14K25 Theta functions and abelian varieties
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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