Torsion points on Jacobians of quotients of Fermat curves and \(p\)-adic soliton theory (Q1340636)

In this paper the author introduces a \(p\)-adic analog of the theory of Sato Grassmannian and of the action of loop groups, inspired by the algebro-geometric and complex-analytic constructions described in the papers of Mumford and Segal-Wilson [cf. \textit{D. Mumford}, Proc. Intl. Symp. Algebraic Geometry, Kyoto 1977, 115-153 (1977; Zbl 0423.14007) and \textit{G. Segal} and \textit{G. Wilson}, Publ. Math., Inst. Hautes Étud. Sci. 61, 5-65 (1985; Zbl 0592.35112)]. As an application of that \(p\)-adic theory, the author proves theorem 1 quoted below. We fix some notations. Let \(\ell\) be an odd prime number and let \(K\) be a field of characteristic zero containing a primitive \(\ell\)-th root of unity \(\zeta\). Let \(X/K\) be the smooth projective model of the plane curve \(y^\ell = x^a(1 - x)^b\), where \(a,b\) are integers such that \(\ell > a \geq b > 0\) and \(a + b = \ell + 1\). Let \(\gamma\) be the unique automorphism of \(X\) such that \(\gamma^* x = x\) and \(\gamma^* y = \zeta y\) and denote by \(\infty\) the unique point of \(X\) at which the coordinate functions \(x\) and \(y\) have poles. Let \(J\) be the Jacobian of \(X\), equipped with the \(\mathbb{Z} [\zeta]\)-module structure for which \(\zeta\) acts by \(\gamma^*\) and let \(\Theta \subset J\) be the set of isomorphism classes of line bundles of degree 0 such that \(h^0 ({\mathcal L} ((g - 1) \infty)) > 0\). Let \(p \equiv 1 \bmod \ell\) be a prime number and let \({\mathfrak p} \subset \mathbb{Z} [\zeta]\) be a prime above \(p\). For each ideal \({\mathfrak a} \subset \mathbb{Z} [\zeta]\), let \(J_{\mathfrak a}\) denote the \({\mathfrak a}\)-torsion subgroup of \(J\). We can finally state the announced Theorem 1. \(\Theta \cap (J_{\mathfrak p} + J_{(1 - \zeta)}) \subseteq J_{(1 - \zeta)}\). In the following, we trace a sketch of the \(p\)-adic \(\tau\)-formalism introduced by the author, by comparing it with the ``classical'' theory on the subject (cf. the papers quoted above). Let \(K\) be a field and let \(X\) be a smooth projective geometrically irreducible curve, equipped with a \(K\)-rational point \(\infty\). Fix a \(K[[1/T]]\)-value point \(N_0\) of \(X\) such that \(N_0 \equiv \infty \pmod {1/T}\) and \(N_0 \not \equiv \infty \pmod {1/T^2}\), and denote by \(N : \text{Spec} K ((1/T)) \to X\) the induced map. A Krichever pair is a couple \(({\mathcal L}, \sigma)\) consisting of a line bundle \({\mathcal L}/X\) and a trivialisation, i.e. a \(K((1/T))\)-linear isomorphism \(\sigma : N^* {\mathcal L} \cong K ((1/T))\), induced by base change from \(\sigma _0 : N^*_0 {\mathcal L} \cong K [[1/T]]\). The fundamental correspondence associates to a Krichever pair a \(K\)-subspace of \(K((1/T))\) in the following way \[ ({\mathcal L}, \sigma) \mapsto \bigl\{ \sigma N^* s : s \in \Gamma (X \backslash \infty, {\mathcal L}) \bigr\} \subseteq K \bigl( (1/T) \bigr). \] The image of this map is contained in the Sato Grassmannian \(\text{Gr}^{alg} (K)\), i.e. in the set of the \(K\)-subspaces \(W \subseteq K((1/T))\) such that the natural map \(W \to K ((1/T))/K [[1/T]]\) has finite kernel and cokernel. The difference between the dimensions of kernel and cokernel of this map is the index of \(W\). There is a natural product of elements in \(\text{Gr}^{alg} (K)\), induced by the product of elements in \(K((1/T))\); this product corresponds to the tensor product of line bundles. Under the above correspondence the set of the isomorphism classes of Krichever pairs is mapped isomorphically to the subgroup \(\text{Gr}_A^{alg} (K)\). Two subspaces \(W\) and \(W'\) in \(\text{Gr}^{alg} (K)\) are homothetic if \(W' = uW\) for some \(u \in K [[1/T]]^\times\). The group law descends under this equivalence and the quotient group \(\text{Gr}_A^{alg} (K)/ \sim\) is isomorphic, by the fundamental correspondence, to the Picard group of \(X\). In particular, the isomorphism classes of line bundles of degree \(n\) correspond to the homothety classes of subspaces of index \(n - g + 1\) and the Theta divisor of \(J\) corresponds to the subset of the homothety classes of subspaces of index \(1 - g\) such that \(\{0\} \neq W \cap T^{g - 1} K [[1/T]]\). Now we pass to the complex-analytic point of view. Suppose \(K = \mathbb{C}\) and change formal Laurent series with convergent Laurent series and suppose that Krichever pairs are strictly defined, i.e. the trivialisation \(\sigma\) of \(N^* {\mathcal L}\) is induced by an analytic trivialisation over a holomorphic embedding of a neighbourhood of the disk \(\{|T |> 1\}\) in the Riemann surface associated to \(X\). To any Laurent series \(\sum a_n T^n\) one can associate the function \(\mathbb{Z} \to \mathbb{C}\), mapping \(n\) to \(a_n\), and one sees that the functions arising from trivialisations of Krichever pairs belong to the Hilbert space of square summable functions, \(\ell^2 (\mathbb{Z})\). Segal and Wilson introduced the Grassmannian \(\text{Gr}^{SW}\) of closed subspaces \(W\) of \(\ell^2 (\mathbb{Z})\) such that the map obtained by composing the inclusion \(W \to \ell^2 (\mathbb{Z})\) with the projection \(\ell^2 (\mathbb{Z}) \to \ell^2 (\mathbb{Z}_{> 0})\) is of Fredholm class, hence it has an index \(i(W)\). the closure in \(\ell^2 (\mathbb{Z})\) of a subspace corresponding to a strictly defined Krichever pair is in \(\text{Gr}^{SW}\), and, given \(W \in \text{Gr}^{SW}\), the intersection \(W^{alg} = W \cap \mathbb{C} ((1/T))\) belongs to \(\text{Gr}^{alg} (\mathbb{C})\) and \(W\) and \(W^{alg}\) have the same index. The Sato \(\tau\)-function is a section of a line bundle over \( \text{Gr}^{SW}\) and its vanishing on a subspace \(W\) is equivalent to the condition \(\{0\} \neq W \cap T^{- i(W)} \mathbb{C} [[1/T]]\). Now let \(K\) be a finite algebraic extension of \(\mathbb{Q}_p\) endowed with the standard extension of the \(p\)-adic absolute value. Consider the ring \[ H(K) = \left\{ \sum_{i \in \mathbb{Z}} a_i T^i :\;a_i \in K, \;\sup_{i \in \mathbb{Z}} |a_i |< \infty,\;\;\lim_{i \to + \infty} |a_i |= 0 \right\} \] endowed with the norm \(|\sum_{i \in \mathbb{Z}} a_i T^i |= \sup_{i \in \mathbb{Z}} |a_i |\) which makes \(H(K)\) a Banach algebra. There is a natural decomposition \(H(K) = H_+ (K) \oplus H_- (K)\), where \(H_+ (K)\) (resp. \(H_- (K))\) is the closed subspace consisting of the series \(\sum_{i \in \mathbb{Z}} a_i T^i\) such that \(a_i = 0\) for \(i \leq 0\) (resp. for \(i > 0)\) and we denote by \(\text{pr}_+ : H(K) \to H_+ (K)\) (resp. \(\text{pr}_- : H(K) \to H_- (K))\) the corresponding projection. The set \(\text{Gr} (K)\) is formed by the (closed) subspaces \(W\) of \(H(K)\) of the form \(W = wH_+ (K)\), where the presentation \(w : H_+ (K) \to H(K)\) is a one-to-one \(K\)-linear map such that, for some integer \(i\), one has \(\text{pr}_+ T^iw = 1 + u\) and \(\text{pr}_- T^iw = v\), where \(u\) and \(v\) are \(K\)-linear, \(|u |\leq 1\), \(|v |\leq 1\) and \(u\) is a uniform limit of bounded operators of finite rank (completely continuous). In particular, the restriction of \(\text{pr}_+\) to \(W\) is a Fredholm operator whose index \(i(W)\) coincides with the integer \(i\) which appears in the presentation of \(W\). If \(W \in \text{Gr} (K)\), the author proves that \(W^{alg} : = W \cap K ((1/T))\) is dense in \(W\) and \(W^{alg} \in \text{Gr }^{alg} (K)\) with \(i(W) = i(W^{alg})\) and he gives a characterization of the image of \(\text{Gr} (K)\) in \(\text{Gr}^{alg} (K)\). In order to define the \(\tau\)-function we have to describe the action of the loop group on \(\text{Gr} (K)\). The group of \(p\)-adic loops is \[ \Gamma (K) = \left\{ h = \sum_{i \in \mathbb{Z}} h_i T^i : |h_0 |= 1,\;\exists \rho\;\bigl( 0 < \rho < 1,\;|h_i |\leq \rho^{\max (i,0)} \forall i \in \mathbb{Z} \bigr) \right\} \subset H (K)^\times, \] and it acts on \(\text{Gr} (K)\) by left multiplication on the presentations. By means of the decomposition \(H(K) = H_+ (K) \oplus H_- (K)\), given a presentation \(w\) of \(W\) and \(h \in \Gamma (K)\), we can write \[ T^{i (W)} w = {1 + u \brack v} \quad \text{and} \quad h = {a \;b \brack c \;d}, \] with \(u\) and \(b\) completely continuous, \(|b |< 1\) and \(u,v,a,c,d\) of norm \(\leq 1\). Consequently the operator \[ hwa^{-1} = T^{- i(W)} \left[ \begin{matrix} 1 + aua^{-1} + bva^{-1} \\ ca^{-1} + cua^{-1} + dv a^{- 1} \end{matrix} \right] \] is a presentation of \(hW \in \text{Gr} (K)\) and \(i(W) = i(hW)\). The \( \tau\) function is defined as \(\tau_w (h) : = \text{det} (1 + u + a^{-1} bv)\), where the determinant is taken in the sense of completely continuous operators [cf. \textit{J.-P. Serre}, Publ. Math., Inst. Hautes Étud. Sci. 12, 69-85 (1962; Zbl 0104.33601)], and the property \(\tau_w (h) = 0 \Leftrightarrow \{0\} \neq hW \cap T^{- i(W)} K[[1/T]]\) holds also in this context.