- Award ID(s):
- 1703182
- NSF-PAR ID:
- 10294557
- Date Published:
- Journal Name:
- Journal of the Institute of Mathematics of Jussieu
- ISSN:
- 1474-7480
- Page Range / eLocation ID:
- 1 to 22
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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null (Ed.)Abstract We prove a local–global compatibility result in the mod $p$ Langlands program for $\mathrm {GL}_2(\mathbf {Q}_{p^f})$ . Namely, given a global residual representation $\bar {r}$ appearing in the mod $p$ cohomology of a Shimura curve that is sufficiently generic at $p$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $p$ completed cohomology is determined by the restrictions of $\bar {r}$ to decomposition groups at $p$ . If these restrictions are moreover semisimple, we show that the $(\varphi ,\Gamma )$ -modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar {r}$ to decomposition groups at $p$ .more » « less
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Abstract Let $$V_*\otimes V\rightarrow {\mathbb {C}}$$ V ∗ ⊗ V → C be a non-degenerate pairing of countable-dimensional complex vector spaces V and $$V_*$$ V ∗ . The Mackey Lie algebra $${\mathfrak {g}}=\mathfrak {gl}^M(V,V_*)$$ g = gl M ( V , V ∗ ) corresponding to this pairing consists of all endomorphisms $$\varphi $$ φ of V for which the space $$V_*$$ V ∗ is stable under the dual endomorphism $$\varphi ^*: V^*\rightarrow V^*$$ φ ∗ : V ∗ → V ∗ . We study the tensor Grothendieck category $${\mathbb {T}}$$ T generated by the $${\mathfrak {g}}$$ g -modules V , $$V_*$$ V ∗ and their algebraic duals $$V^*$$ V ∗ and $$V^*_*$$ V ∗ ∗ . The category $${{\mathbb {T}}}$$ T is an analogue of categories considered in prior literature, the main difference being that the trivial module $${\mathbb {C}}$$ C is no longer injective in $${\mathbb {T}}$$ T . We describe the injective hull I of $${\mathbb {C}}$$ C in $${\mathbb {T}}$$ T , and show that the category $${\mathbb {T}}$$ T is Koszul. In addition, we prove that I is endowed with a natural structure of commutative algebra. We then define another category $$_I{\mathbb {T}}$$ I T of objects in $${\mathbb {T}}$$ T which are free as I -modules. Our main result is that the category $${}_I{\mathbb {T}}$$ I T is also Koszul, and moreover that $${}_I{\mathbb {T}}$$ I T is universal among abelian $${\mathbb {C}}$$ C -linear tensor categories generated by two objects X , Y with fixed subobjects $$X'\hookrightarrow X$$ X ′ ↪ X , $$Y'\hookrightarrow Y$$ Y ′ ↪ Y and a pairing $$X\otimes Y\rightarrow {\mathbf{1 }}$$ X ⊗ Y → 1 where 1 is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $${\mathbb {T}}$$ T and $${}_I{\mathbb {T}}$$ I T .more » « less
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Abstract Let
denote the matrix multiplication tensor (and write$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$ ), and let$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$ denote the determinant polynomial considered as a tensor. For a tensor$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$ T , let denote its border rank. We (i) give the first hand-checkable algebraic proof that$\underline {\mathbf {R}}(T)$ , (ii) prove$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$ and$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$ , where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$ , (iii) prove$M_{\langle 2\rangle }$ , (iv) prove$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$ , improving the previous lower bound of$\underline {\mathbf {R}}(\operatorname {det}_3)=17$ , (v) prove$12$ for all$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$ , where previously only$\mathbf {n}\geq 25$ was known, as well as lower bounds for$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$ , and (vi) prove$4\leq \mathbf {n}\leq 25$ for all$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$ , where previously only$\mathbf {n} \ge 18$ was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors.$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$ The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, called
border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorT and an integerr , in a finite number of steps, either outputs that there is no border rankr decomposition forT or produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenT has a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory. -
We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $p$ . This is a generalization to $\text{GL}_{3}$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $n$ -dimensional Galois representations’, Duke Math. J. 149 (1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math. 212 (1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $\text{GL}_{3}(\mathbb{F}_{q})$ .more » « less
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Let ρ ¯ : G Q → GSp 4 ( F 3 ) \overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3) be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider ρ ¯ \overline {\rho } to be the Galois representation associated to the 3 3 -torsion of a principally polarized abelian surface A / Q A/\mathbf {Q} . We prove that the moduli space A 2 ( ρ ¯ ) \mathcal {A}_2(\overline {\rho }) of principally polarized abelian surfaces B / Q B/\mathbf {Q} admitting a symplectic isomorphism B [ 3 ] ≃ ρ ¯ B[3] \simeq \overline {\rho } of Galois representations is never rational over Q \mathbf {Q} when ρ ¯ \overline {\rho } is surjective, even though it is both rational over C \mathbf {C} and unirational over Q \mathbf {Q} via a map of degree 6 6 .more » « less