In the 1970's, Serre exploited congruences between q-expansion coefficients of Eisenstein series to produce p-adic families of Eisenstein series and, in turn, p-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to p-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on q-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack q-expansions when the signature is of the form (a,b), a≠b. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a p-adic measure taking values in the space of p-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit p-adic family of Eisenstein series. One application of our results is to the recently completed construction of p-adic L-functions for unitary groups by the first named author, Harris, Li, and Skinner.
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-ADIC -FUNCTIONS FOR UNITARY GROUPS
This paper completes the construction of $p$ -adic $L$ -functions for unitary groups. More precisely, in Harris, Li and Skinner [‘ $p$ -adic $L$ -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math. Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $p$ -adic $L$ -functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and $p$ -adic differential operators [Eischen, ‘A $p$ -adic Eisenstein measure for unitary groups’, J. Reine Angew. Math. 699 (2015), 111–142; ‘ $p$ -adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble) 62 (1) (2012), 177–243], Part II of the present paper provides the calculations of local $\unicode[STIX]{x1D701}$ -integrals occurring in the Euler product (including at $p$ ). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.
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- Award ID(s):
- 1751281
- NSF-PAR ID:
- 10148629
- Date Published:
- Journal Name:
- Forum of Mathematics, Pi
- Volume:
- 8
- ISSN:
- 2050-5086
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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In the 1970's, Serre exploited congruences between q-expansion coefficients of Eisenstein series to produce p-adic families of Eisenstein series and, in turn, p-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to p-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on q-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack q-expansions when the signature is of the form (a,b), a≠b. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a p-adic measure taking values in the space of p-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit p-adic family of Eisenstein series. One application of our results is to the recently completed construction of p-adic L-functions for unitary groups by the first named author, Harris, Li, and Skinner.more » « less
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null (Ed.)Following a suggestion of Peter Scholze, we construct an action of G m ^ \widehat {\mathbb {G}_m} on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p p -adic modular curve whose ring of functions is Serre’s space of p p -adic modular functions. This action is a local, p p -adic analog of a global, archimedean action of the circle group S 1 S^1 on the lattice-unstable locus of the modular curve over C \mathbb {C} . To construct the G m ^ \widehat {\mathbb {G}_m} -action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates q q ; along the way we also prove a natural generalization of Dwork’s equation τ = log q \tau =\log q for extensions of Q p / Z p \mathbb {Q}_p/\mathbb {Z}_p by μ p ∞ \mu _{p^\infty } valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of G m ^ \widehat {\mathbb {G}_m} integrates the differential operator θ \theta coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p p -adic L L -functions.more » « less
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Abstract We prove that
$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$ D -dimensional lattice withn qudits are approximatet -designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was$${{\,\textrm{poly}\,}}(t)\cdot n$$ $$D=1$$ $${{\,\mathrm{\textsf{PH}}\,}}$$ $$\#{\textsf{P}}$$ $$O(\sqrt{n})$$ t -designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size$$O(n\ln ^2 n)$$ $$O(\ln ^3 n)$$ $$\Omega (n \ln n)$$ $$O(\ln n \ln \ln n)$$ $$O(n \ln n \ln \ln n)$$ -
null (Ed.)Abstract Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $p$-adic and $\textrm{mod}\; p$ modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: (1) the analytic continuation at unramified primes $p$ to the whole Shimura variety of the $\textrm{mod}\; p$ reduction of $p$-adic Maass–Shimura operators a priori defined only over the $\mu $-ordinary locus, and (2) the construction of new $\textrm{mod}\; p$ theta operators that do not arise as the $\textrm{mod}\; p$ reduction of Maass–Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fmp.2020.4
