More Like this

1. MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL

   https://doi.org/10.1017/S1474748020000225  

   Le, Daniel ; Morra, Stefano ; Schraen, Benjamin ( May 2020 , Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $v$ above $p$ . Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We describe the $\mathfrak{m}$ -torsion in the $\text{mod}\,p$ cohomology of Shimura curves with full congruence level at $v$ as a $\text{GL}_{2}(k_{v})$ -representation. In particular, it only depends on $\overline{r}|_{I_{F_{v}}}$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $\text{GL}_{2}(\mathbf{F}_{q})$ -projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math.   200 (1) (2015), 1–96]. 

   more » « less
2. -ADIC -FUNCTIONS FOR UNITARY GROUPS

   https://doi.org/10.1017/fmp.2020.4  

   EISCHEN, ELLEN ; HARRIS, MICHAEL ; LI, JIANSHU ; SKINNER, CHRISTOPHER ( January 2020 , Forum of Mathematics, Pi)

   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. 

   more » « less
3. A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

   https://doi.org/10.1090/bproc/141  

   Lang, Jaclyn ; Wake, Preston ( October 2022 , Proceedings of the American Mathematical Society, Series B)

   We show that for primes$N, p \geq 5$with$N \equiv -1 \bmod p$, the class number of$\mathbb {Q}(N^{1/p})$is divisible by$p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when$N \equiv -1 \bmod p$, there is always a cusp form of weight$2$and level$\Gamma _0(N^2)$whose$\ell$th Fourier coefficient is congruent to$\ell + 1$modulo a prime above$p$, for all primes$\ell$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree-$p$extension of$\mathbb {Q}(N^{1/p})$.

    

   more » « less
4. Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

   https://doi.org/10.1002/mana.201700415  

   Vasiu, Adrian ( October 2020 , Mathematische Nachrichten)

   We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0,p). As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic (0,p) of integral canonical models of projective Shimura varieties of Hodge type with respect to h‐hyperspecial subgroups as pro‐étale covers of Néron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.

    

   more » « less
5. Regular integral models for Shimura varieties of orthogonal type

   https://doi.org/10.1112/S0010437X22007370  

   Pappas, G. ; Zachos, I. ( April 2022 , Compositio Mathematica)

   We consider Shimura varieties for orthogonal or spin groups acting on hermitian symmetric domains of type IV. We give regular $p$ -adic integral models for these varieties over odd primes $p$ at which the level subgroup is the connected stabilizer of a vertex lattice in the orthogonal space. Our construction is obtained by combining results of Kisin and the first author with an explicit presentation and resolution of a corresponding local model. 

   more » « less