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Non-abelian \(p\)-adic \(L\)-functions and Eisenstein series of unitary groups – the CM method. (\(L\)-fonctions \(p\)-adiques non-abéliennes et série d’Eisenstein pour les groupes unitaires – la méthode CM.) (English. French summary) Zbl 1396.11124

The equivariant (abelian or not) main conjecture of Iwasawa theory predicts a deep relationship between an analytic object (a \(p\)-adic \(L\)-function) and an algebraic object (a Selmer group or complex attached to a motive over a \(p\)-adic Lie extension). For totally real \(p\)-adic Lie extensions and the Tate motive, the EMC has been proved in the abelian case by Wiles, and in the non-abelian case by Ritter and Weiss, and independently by Kakde. For other motives, one of the main difficulties lies in the construction of non-abelian \(p\)-adic \(L\)-functions. The main aim of the present article is to tackle this problem for motives (in a loose sense) whose classical \(L\)-functions can be studied through \(L\)-functions of automorphic representations of definite unitary groups, and in certain cases, to prove for them the so-called “torsion congruences” previously shown by J. Ritter and A. R. Weiss for the Tate motive [Math. Res. Lett. 15, No. 4, 715–725 (2008; Zbl 1158.11047)], and by the author (under certain conditions) for the motive associated to a CM elliptic curve [Int. J. Number Theory 7, No. 7, 1883–1934 (2011; Zbl 1279.11107)].
To describe these congruences, we must introduce some notations and definitions. Fix an odd prime number \(p\), and let \(F'/F\) be a totally real Galois extension with Galois group \(\Gamma\) of order \(p\), unramified outside \(p\). Write \(G_F:=\mathrm{Gal}(F(p^\infty)/F)\), where \(F(p^\infty)\) is the maximal abelian extension of \(F\) unramified outside \(p\). The assumption on the ramification of \(F'/F\) implies the existence of a transfer map \(\text{ver}: G_F\to G_F\), which also induces a map \(\text{ver}: \mathbb Z_p[[G_F]]\to \mathbb Z_p[[G_F]]\). For a motive \(M/F\) in a loose sense, i.e., defined by its usual realizations and the compatibilities between them, it is conjectured, under some assumptions on the critical values of \(M\) and some ordinary assumptions at \(p\), that there exists an element \(\mu_F\in \mathbb Z_p[[G_F]]\) which interpolates the critical values of \(M/F\) twisted by characters of \(G_F\). Write \(\mu_F\) for the element in \(\mathbb Z_p[[G_{F'}]]\) associated to \(M/F'\), the base change of \(M/F\) to \(F'\). Then the so-called torsion congruences read \[ \text{ver}(\mu_F)\equiv \mu_{F'}\pmod T, \]
where \(T\) is the trace ideal in \(\mathbb Z_p[[G_{F'}]]^\Gamma\).
To describe the author’s results towards these congruences, we need to introduce further a totally imaginary quadratic extension \(K\) of \(F\) such that all \(p\)-primes of \(F\) are split in \(K\), and \(K'=F'\cdot K\). Fix an embedding \(i_p: \overline{\mathbb Q} \hookrightarrow \mathbb C_p\). The splitting of the \(p\)-primes in \(K\) implies the existence of an ordinary CM type \(\Sigma\) of \(K\) in the sense of Katz (i.e., such that for any \(\sigma\) in \(\Sigma\) and \(\tau\) in the conjugate of \(\Sigma\), the \(p\)-adic valuations induced from \(i_p\cot \sigma\) and \(i_p\cdot \tau\) are inequivalent). Let \(\phi\) be a Hecke character of \(K\) with infinite type \(\Sigma\) for some integer \(\geq 1\). Write \(M(\phi)/F\) for the motive over \(F\) obtained by Weil restriction to \(F\) from the rank 1 motive over \(K\) associated to \(\psi\). For any finite character \(\psi\) of \(G_F\), we have
\[ L(M(\psi)\otimes\psi,s)=L(\psi \tilde{\chi},s), \]
where \(\tilde{\chi}= \chi\cdot N_{K'/K}\) (it is the base change of \(\psi\) from \(K\) to \(K'\), of infinite type \(-k\Sigma')\). Consider then a Hermitian space \((W,\theta)\) over \(K\), where \(W\) is a \(K\)-vector space of dimension \(n\), and \(\theta\) is a non-degenerate Hermitian form on \(W\). Assume that for any \(\sigma\) in \(\Sigma\), the signature of the form induced by \(\theta\) on \(W\otimes_{K,\sigma}\mathbb C\) is the same; this implies in particular that the splitting of the \(p\)-primes in \(K\) is the usual ordinary condition. Write \(U(\theta)\) for the corresponding unitary group. Consider now a motive \(M(\pi)/K\) such that there exists an automorphic representation \(\pi\)unitary group \(U((\theta)\mathbb A_F)\) with the property that the \(L\)-function \(L(M(\pi)/K,s)\) is equal to \(L(\psi,s)\). Assume the following:
(1) The \(p\)-adic realiztions of \(M(\pi)\) and \(M(\psi)\) have \(\mathbb Z_p\)-coefficients.
(2) The infinite type of the representation \(\pi\) is of parallel scalar weight \(\ell\).
(3) \(k+2\ell\geq n\).
Then, for \(n=1\), the author constructs a \(p\)-adic measure \(\mu_{M(\psi)/F}\) which interpolates values of the \(L\)-function associated to \(\psi\) twisted by finite Hecke characters of \(K\). This is very similar to the measure of N. M. Katz [Invent. Math. 49, No. 3, 199–297 (1978; Zbl 0417.12003)] and H. Hida and J. Tilouine [Ann. Sci. Éc. Norm. Supér. (4) 26 , No. 2, 189–259 (1993; Zbl 0778.11061)], but with differences on some normalizing factors as well as on the Euler factors (which are removed here).
Theorem 1: For \(n=1\), the torsion congruences hold true, i.e.,
\[ \text{ver}\left(\mu_{M(\psi)/F}\right) \equiv \mu_{M(\psi')/F'} \pmod T. \]
The case \(n=2\) requires control of the base change of the automorphic representation, which in principle is a hard problem. The author circumvents it by imposing an additional hypothesis (H) which ensures the existence of a \(\mathbb Q_p\)-valued modular form \(f_\pi\) of \(U(\theta)/F\) which is an eigenform for all Hecke operators away from a large enough integral ideal \(\mathfrak c\) of \(K\) which contains the conductor of \(\pi\), and \(f_\pi\) verifies certain technical conditions (too technical to be recalled here). A family of examples verifying (H) can be provided.
Theorem 2: For \(n=2\) under hypothesis (H), one has the congruence
\[ \Omega^{2\ell}, \langle f_\pi, \check f_\pi\rangle, \text{ver}\binom{(f_\pi)}{\mu_{(\phi,\psi)}} \equiv \langle f_\pi, \hat f_\pi\rangle \mu^{(f_{\pi'})}(\pi',\psi')\pmod T. \]
Here the motive \(M(\pi,\psi)/F\)is obtained by Weil restriction from the motive \(\psi\otimes M(\pi)/K\), where \(\psi\) is thought of as the rank 1 motive over \(K\) associated to the Hecke character \(\psi\), and for a character \(\chi\) of \(G_F\),
\[ L(M(\pi,\psi)/F,\chi,s) = L(\pi, \psi\tilde\chi,s),\quad\text{with}\;\tilde\chi=\chi\circ N_{K/F}. \] The author starts from the Eisenstein measure of M. H. Harris, J. S. Li and C. M. Skinner [in: Automorphic representations, L-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ. 11, 225–255 (2005; Zbl 1103.11017)] to construct the \(p\)-adic measure \(\mu^{(f_\pi)}_{(\pi,\psi)}\) which interpolates critical values (and their twists) of the \(L\)-functions which come into play (note that this construction encompasses the previous construction of \(\mu_{M(\psi)/F})\). The standard normalized Peterson product of \(\pi\) is denoted \(\langle\cdot,\cdot\rangle\). The factor \(\Omega\in\mathbb Z_p^*\) is, roughly speaking, the quotient of two \(p\)-adic periods. It can be removed modulo an extra condition.
Finally the author can get rid of hypothesis (H) by “averaging” the torsion congruences. More precisely: write \(G=U(\theta)/F\) and \(\text{Rep}(G,\mathfrak g)= \) the set of automorphic representations of \(G\) of a conductor contained in \(\mathfrak c\) and of parallel weight \(\ell\); one can define a certain (technical) compact subgroup \(D(\mathfrak c)\subset G(\mathbb A_F,f)\) and fix an orthogonal basis \(\{f_j\}\) consisting of Hermitian forms of \(G\) for the congruence subgroup \(D(\mathfrak c)\), of parallel weight \(\ell\), and which are eigenforms for all relative prime to \(\mathfrak c\) Hecke operators; let \(\mathcal B_K\) be the finite set of double cases \(G(F)\subset G(\mathbb A_F,f)/D(\mathfrak c)\); for \(a,b\in\mathcal B_K\), define the twisted measure
\[ \mu_{F,(a,b)}:= \sum_j \frac{f_j(a)f_j(b)}{[ f_j,f_j]} \mu^{(f_j)}_{(\pi_j,\psi)} \in \mathbb Z_p[[G_F]], \]
where \(f_j\) is associated to some \(\pi_j\in \text{Rep}(G,\mathfrak c)\).
Theorem 3 (without assuming (H)):
(1) For any \(\mathbb Z_p\)-valued locally constant function \(\varepsilon\) on \(\mathbb Z_p[[G_{F'}]]\) which is fixed by \(\gamma\),
\[ \Omega^{2\ell} \int_{G_F} \varepsilon\circ\text{ver}\,d\mu_{F,(a,b)} \equiv \int_{G_F} \varepsilon \,d\mu_{F',(a,b)} \pmod p. \]
(2) If \(F'/F\) is unramified at \(p\), there is a constant \(c(a,b)\in\mathbb Z_p\) such that
\[ c(a,b)\Omega^{2\ell} \text{ver}\,\mu_{F,(a,b)}\equiv c(a,b)\mu_{F',(a,b)} \pmod T, \]
i.e., the torsion congruences hold for all twisted normalized measures \[ c(a,b)\mu_{F,(a,b)}\quad\text{and}\quad c(a,b)\mu_{F',(a,b)}, \quad a,b\in\mathcal B_K. \]

MSC:

11R23 Iwasawa theory
11F55 Other groups and their modular and automorphic forms (several variables)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
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References:

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