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1. Differential operators and families of automorphic forms on Unitary groups of arbitrary signature

   Eischen, Ellen; Fintzen, Jessica; Mantovan, Elena; Varma, Ila (June 2018, Documenta mathematica)

   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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2. Differential operators and families of automorphic forms on Unitary groups of arbitrary signature

   https://doi.org/10.25537/dm.2018v23.445-495  

   Eischen, Ellen; Fintzen, Jessica; Mantovan, Elena; Varma, Ila (June 2018, Documenta mathematica)

   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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3. A unipotent circle action on 𝑝-adic modular forms

   https://doi.org/10.1090/btran/52  

   Howe, Sean (November 2020, Transactions of the American Mathematical Society, Series B)

   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. 

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4. Entire Theta Operators at Unramified Primes

   https://doi.org/10.1093/imrn/rnab190  

   Eischen, E; Mantovan, E (July 2021, International Mathematics Research Notices)

   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. 

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5. Spectral decomposition and Siegel–Veech transforms for strata: the case of marked tori

   https://doi.org/10.4171/JST/563  

   Athreya, Jayadev S; Lagacé, Jean; Möller, Martin; Raum, Martin (May 2025, Journal of Spectral Theory)

   Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel–Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemann surfaces. This space carries actions of the foliated Laplacian derived from the \mathrm{SL}_{2}(\mathbb{R})-action as well as various differential operators related to relative period translations.In the paper we give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows to employ techniques from representation theory and global analysis. Even for this simple stratum, exhibiting all Siegel–Veech transforms requires novel configurations of saddle connections. We also show that the continuous spectrum of the foliated Laplacian is much larger than the space of Siegel–Veech transforms, as opposed to the case of the modular curve. This defect can be remedied by using instead a compound Laplacian involving relative period translations. 

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