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A reciprocity map and the two-variable \(p\)-adic \(L\)-function. (English) Zbl 1248.11085
From the author’s introduction: “The principal theme of this article is that special elements in the Galois cohomology of a cyclotomic field should correspond to special elements in the quotient of the homology group of a modular curve by an Eisenstein ideal. The elements on the Galois side of the picture arise as cup products of units in our cyclotomic field, while the elements on the modular side arise in alternate forms of our conjecture from Manin symbols and \(p\)-adic \(L\)-values of cusp forms that satisfy congruences with Eisenstein series at primes over \(p.\) We can also understand this as a comparison between objects that interpolate these elements: the value of a reciprocity map on a particular norm compatible sequence of \(p\)-units and an object giving rise to a two-variable \(p\)-adic \(L\)-function, taken modulo an Eisenstein ideal.”
Actually, the core of this paper is focused on the statements of the various forms of the author’s conjecture and the proofs of their equivalence. Let us try and give an overview (not too technically precise) of the constructions involved. Choose a prime \(p \geq 5\) and a positive integer \(N\) prime to \(p\) not dividing the value \(\varphi (N)\) of Euler’s phi function. For any integer \(r \geq 0,\) write \(F_r = {\mathbb Q} (\mu_{N p^r})\) and \(F_\infty = \displaystyle\mathop\bigcup_{r \geq 0} F_r = {\mathbb Q} (\mu_{N p^\infty}).\) Denote by \(S\) the set of primes over \(N p\) and any real places of any given algebraic extension \(K/{\mathbb Q},\) and \(G_{K, S}\) the Galois group of the maximal unramified outside \(S\) extension of \(K.\) Let us introduce the following objects:
On the Galois side: Form the cup product \(H^1_{cont} \bigl( G_{F_r, S}, {\mathbb Z}_p (1)\bigl)^{\otimes 2} \longrightarrow H^2_{cont} \Bigl( G_{F_{r, S}}, {\mathbb Z}_p (2) \Bigl).\) Denote by \((\ldotp, \ldotp)^0_{F_{r, S}}\) the projection of the resulting pairing on \(S\)-units of \(F_r\) to the sum of odd, primitive eigenspaces of the second cohomology group under a \((-1)\)-twist of the standard action of \(({\mathbb Z}/N p\;{\mathbb Z})^\times.\) We are interested in the values \(\bigl( 1 - \zeta^i_{N p^r}, 1-\zeta^j_{N p^r} \bigl)^0_{F_{r, S}}\) of this pairing for \(i, j \in {\mathbb Z}\) non zero mod \(N p^r\) with \((i, j, N p) = 1.\)
Take \(K = F_\infty\) and let \({\mathfrak X}_K\) be the maximal abelian pro-\(p\)-quotient of \(G_{K, S},\) with trivial action of \(G_{K, S}.\) The exact sequence \(1 \to {\mathfrak X}_K \to {\mathcal J} \to {\mathbb Z}_p \to 0\) of \({\mathbb Z}_p [ [ G_{K, S} ] ] \)-modules, where \({\mathcal J}\) is determined by the cocycle proj: \(G_{K, S} \to {\mathfrak X}_K,\) yields a coboundary map \(\Psi_K\:\;H^1_{Iw} (G_{K, S}, {\mathbb Z}_p (1)) \to H^2_{Iw} (G_{K, S}, {\mathbb Z}_p (1)) \displaystyle\mathop\otimes_{{\mathbb Z}_p} {\mathfrak X}_K,\) where \(H^i_{Iw} (\ldotp)\) denotes as usual \(\displaystyle\mathop{\lim_{\displaystyle\mathop\leftarrow_r}} \;H^i_{cont} (\ldotp).\) The odd, primitive part \(X^0_K\) of the maximal unramified quotient \(X_K\) of \({\mathfrak X}_K\) is known to be isomorphic to the odd, primitive part of \(H^2_{Iw} (G_{K, S}, {\mathbb Z}_p 1)).\) Define \(\Psi^0_K\) (the so-called “reciprocity map”) by composing with projection to \(X^0_K \displaystyle\mathop\otimes_{{\mathbb Z}_p} {\mathfrak X}^-_K.\) We are interested in the value \(\Psi^0_K (1-\zeta)\) on the norm compatible sequence \(1-\zeta = (1-\zeta_{N p^r})_r.\)
Finally, let us consider cup products with twisted coefficients. Let \(\omega\) denote the Teichmüller character and \(\kappa = \chi_{cyc}\;\omega^{-1}.\) Let \({\mathcal O}_{N p^r}\) be the extension of \({\mathbb Z}_p\) generated by the values of all \(\overline {\mathbb Q}_p\)-valued characters of \(\bigl( {\mathbb Z}_{/N p^r {\mathbb Z}} \bigl)^\times.\) For any even \(p\)-adic character \(\psi\) of \(\bigl( {\mathbb Z}_{N p^s {\mathbb Z}} \bigl)^\times\) \((s \geq 1)\) and \(t \in {\mathbb Z}_p,\) define
\[ \alpha^\psi_t = \displaystyle \lim_{r \to \infty} \displaystyle\prod^{N p^r -1}_{{i=1}\atop (i, N p) = 1}\bigl( 1 - \zeta^i_{N p^r} \bigl)^{\psi \,\kappa^{t-1}(i)} \in H^1_{cont} \bigl( G_{{\mathbb Q}, S}, {\mathcal O}_{N p^s} (\kappa^t \omega \psi) \bigl). \]
For \(k \in {\mathbb Z}_p\) and \({\theta}\) an odd character of \(\bigl( {\mathbb Z}_{/N p^s {\mathbb Z}}\bigl)^\times\) such that the restriction of \({\theta}\) to \(\bigl( {\mathbb Z}_{/N p {\mathbb Z}}\bigl)^\times\) is primitive, we are interested in the cup products \(\alpha^\psi_t \cup \alpha_{k-1}^{\theta \psi^{-1} \omega^{-1}} \in H^2 \left(G_{{\mathbb Q},S}, {\mathcal O}_{N p^s} (\kappa^t \omega \theta)\right)\).
On the modular side:
Consider the singular homology group \(H_1 (X_1 (N\, p^r)\;;\;{\mathbb Z}_p),\) which is a module for a cuspidal Hecke algebra with the standard action on homology. The ordinary part, that is the submodule on which the Hecke operator \(U_p\) is invertible, contains symbols \(\xi_r (i\,:\, j)\) (for \(i,j \in {\mathbb Z}\) with \((i, j, N p = 1)\) arising from classes of paths between cusps in the upper half-plane. Inside the part of the cuspidal \({\mathbb Z}_p\)-algebra that is ordinary and primitive under a certain twisted action of the diamond operators, one has the Eisenstein ideal \(I_r\) generated by projections of elements of the form \(T_\ell - 1 - \ell \langle \ell \rangle,\) with \(\ell\) prime and \(\ell |\!\!/ N p,\) along with \(U_\ell - 1\) for \(\ell | N p.\) Let \(Y_r\) denote the localization of \(H_1 (X_1 (N p^r)\;;\;{\mathbb Z}_p)\) (identified with \(H^1_{\text{ét}} (X_1 (N p^r)_{/ \overline{\mathbb Q}}\;;\;{\mathbb Z}_p)\) at the ideal \({\mathcal M}_r\) generated by \(I_{r}, p\) and \(\langle 1 + p \rangle - 1.\) We are interested in the projection \(\overline \xi_r (i\,:\, j)\) of \(\xi_r (i\,:\, j)\) to \(Y^-_r / I_r Y^-_r.\) Starting from the symbols \(\xi_r (i\,:\, 1),\) one can define two-variable \(p\)-adic \(L\)-functions, more precisely sequences of Mazur-Tate elements which interpolate such \(L\)-functions. The \(L\)-function \({\mathcal L}_N\) defined by \({\mathcal L}_N = \displaystyle\mathop{\lim_{\displaystyle\mathop\leftarrow_r}} \displaystyle\prod^{N p^r -1}_{{i=1}\atop (i, N p) =1}\;U^{-1}_p \xi_r (i\,:\, 1) \otimes [i]_r,\) where \([i]_r\) denotes the element of \({\mathbb Z}_p [({\mathbb Z}/N p^r {\mathbb Z})^\times]\) corresponding to \(i,\) is essentially the two-variable \(p\)-adic \(L\)-function of Mazur and Kitagawa (see K. Kitagawa [Contemp. Math. 165, 81–110 (1994; Zbl 0841.11028)]). We are interested in the modified function \({\mathcal L}^\bullet_N = \displaystyle\mathop{\lim_{\displaystyle\mathop\leftarrow_r}}\;\displaystyle\prod^{N p^r -1}_{i=1} U^{-1}_p \xi_r (i\,:\,1) \otimes [i]_r\) (where \(({\mathbb Z}/ N p^r {\mathbb Z})^\times\) is replaced by \(\bigl({\mathbb Z}/N p^r {\mathbb Z}\bigl)^\bullet = {\mathbb Z}/N p^r {\mathbb Z} \backslash (0)\)) and its projection \(\overline{{\mathcal L}^\bullet_N}\) to \({\mathcal Y}_N^- / {\mathfrak I}\, {\mathcal Y}_N^- \displaystyle\mathop\otimes_{{\mathbb Z}_p} \left(\bigwedge^\bullet_N\right)^-\). Here \({\mathcal Y}_N = \displaystyle\lim_\leftarrow Y_r,\) \({\mathfrak I} = \displaystyle\lim_\leftarrow I_r\) and \(\bigwedge^\bullet_N = \displaystyle\lim_\leftarrow {\mathbb Z}_p [({\mathbb Z}_{/ N p^r {\mathbb Z}})^\bullet].\) Finally, the special values of \({\mathfrak L}_N\) are defined starting from \(\displaystyle\mathop{\lim_{\displaystyle\mathop\leftarrow_r}} \displaystyle\prod^{N p^r -1}_{{i=1}\atop (i, N p) =1} \psi K^{t-1} (i) \xi_n (i: 1)\); the image of this limit in the maximal quotient on which each diamond operator \(\langle j \rangle\) acts as \(\theta \omega^{-1} \kappa^{k-2} (j)\) is denoted \(L_p (\xi, \omega \theta, k, \psi, t),\) because it interpolates the values at the given \(t \in {\mathbb Z}_p\) of the \(L_p\)-functions with character \(\psi\) of the ordinary cusp forms of weight \(k,\) level \(N p^s,\) and character \(\theta \omega^{-1}\) (here \(\kappa, \psi, \theta\) have the same-meaning as on the Galois side). We are interested in the reduction \(\overline{L_p (\xi, \omega \theta, k, \psi, t)}\) modulo the Eisenstein ideal of weight \(k\) and character \(\theta \omega^{-1}.\)
To relate the Galois side and the modular side, the author constructs two maps. The first map, \(\phi_1: X^0_K \to {\mathcal Y}^-_N / {\mathfrak I} {\mathcal Y}^-_N,\) arises from the Galois action of \(G_{K, S}\) on \({\mathcal Y}_N\) ; the second map, \(\phi_2: {\mathfrak X}^-_K \to {\mathbb Z}_p [[ {\mathbb Z}^\bullet_{p, N}]]^-,\) is determined by the action of \({\mathfrak X}^-_K\) on \(p\)-power roots of cyclotomic \(N p\)-units. The three main equivalent versions of the author’s conjecture then state:
The Tate twist of \(\phi_1\) by \({\mathbb Z}_p(1)\) sends \(\bigl( 1 - \zeta_{N p^r}, 1 - \zeta^j_{N p^r}\bigl)^0_{F_r, S}\) to \(\overline \xi_r (i: j)\).
A certain twist of \(\phi_1,\)taking appropriate quotients, sends \(\Psi^0_K (1-\zeta)\) to \(\overline{{\mathcal L}^\bullet_n}\).
\(\phi_1 \otimes \phi_2\) sends \(\alpha^\psi_t \cup \alpha_{k-t}^{\theta \psi^{-1} \omega^{-1}}\) to \(\overline{L_p (\xi, \omega \theta, k, \psi , t)}\).
At present, the most convincing evidence is a proof of a particular specialization of the conjecture R. T. Sharifi [Duke Math. J. 137, No. 1, 63–101 (2007; Zbl 1131.11068)]. Note also that the new conjecture contains a large part of a former one in the image of the cup product [W. G. McCallum and R. T. Sharifi, Duke Math. J. 120, 269–310 (2003; Zbl 1047.11106)]. An analogue of a variant concerning an “inverse” map of the one in (1) (for \(N = 1\) and assuming Vandiver’s conjecture) can also be found in [C. Buisioc, Trans. Am. Math. Soc. 360, No. 11, 5999–6015 (2008; Zbl 1225.11067)].

11R23 Iwasawa theory
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11R34 Galois cohomology
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