Similitude Group Research Articles

Arthur Packets for Quasisplit GSp(2n) and GO(2n) Over a p-Adic Field

Abstract We construct the Arthur packets for symplectic and even orthogonal similitude groups over a $p$-adic field and show that they are stable and satisfy the twisted endoscopic character relations.

Higher level cusp forms on exceptional group of type E7

By using new techniques with the degenerate Whittaker functions developed by Ikeda and Yamana, we construct a higher-level cusp form on GE7,3 (exceptional similitude group of type E7), called Ikeda type lift, from any Hecke cusp form whose corresponding automorphic representation has no supercuspidal local components. This generalizes our earlier results on level-1 forms. But there is a new phenomenon in higher levels; we can handle cusp forms with nontrivial central characters. We also compute the degree-133 adjoint L-function of the Ikeda type lift.

The Kottwitz conjecture for unitary PEL-type Rapoport–Zink spaces

Abstract In this paper we study the cohomology of PEL-type Rapoport–Zink spaces associated to unramified unitary similitude groups over ℚ p {\operatorname{\mathbb{Q}}_{p}} in an odd number of variables. We extend the results of Kaletha–Minguez–Shin–White and Mok to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapoport–Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider.

Hodge classes and the Jacquet–Langlands correspondence

Abstract We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in $\ell $ -adic cohomology is Galois invariant for all $\ell $ .

Picard modular forms and the cohomology of local systems on a Picard modular surface

We formulate a detailed conjectural Eichler–Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes \operatorname{GU}(2,1, \mathbb{Q}(\sqrt{-3})) . The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula. Along the way we prove new results on characteristic polynomials of Frobenius acting on the first cohomology group of cyclic triple covers of any genus, dimension formulas for spaces of Picard modular forms and formulas for the numerical Euler characteristics of the local systems.

Pseudo-differential operators, Wigner transform and Weyl transform on the Similitude group, [formula omitted

In this paper we study pseudo-differential operators with operator valued symbols on the Similitude group, SIM(2). We characterize the unique infinite-dimensional, unitary, irreducible representations of SIM(2) and using that the Fourier inversion formula is given. The operator valued Wigner transform and the Weyl transform are also studied here. Furthermore we also obtain a necessary and sufficient condition on the symbols for which the pseudo-differential operators SIM(2) are Hilbert-Schmidt and trace class operators. Finally, we also give the trace formula for these operators.

Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations

We show that equivariant tilting modules over equivariant algebras induce equivalences of derived factorization categories. As an application, we show that the derived category of a noncommutative resolution of a linear section of a Pfaffian variety is equivalent to the derived factorization category of a noncommutative gauged Landau-Ginzburg model $(\Lambda,\chi, w)^{\mathbb{G}_m}$, where $\Lambda$ is a noncommutative resolution of the quotient singularity $W/\operatorname{GSp}(Q)$ arising from a certain representation $W$ of the symplectic similitude group $\operatorname{GSp}(Q)$ of a symplectic vector space $Q$.

IHARA LEMMA AND LEVEL RAISING IN HIGHER DIMENSION

AbstractA key ingredient in the Taylor–Wiles proof of Fermat’s last theorem is the classical Ihara lemma, which is used to raise the modularity property between some congruent Galois representations. In their work on Sato and Tate, Clozel, Harris and Taylor proposed a generalisation of the Ihara lemma in higher dimension for some similitude groups. The main aim of this paper is to prove some new instances of this generalised Ihara lemma by considering some particular non-pseudo-Eisenstein maximal ideals of unramified Hecke algebras. As a consequence, we prove a level-raising statement.

On supersingular loci of Shimura varieties for quaternionic unitary groups of degree 2

We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree 2 over a ramified odd prime p if the level at p is given by a special maximal compact open subgroup. More precisely, we show that such a locus is purely 2-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport–Zink space for the quaternionic unitary similitude group of degree 2, with a special parahoric level. We prove that such a scheme is purely 2-dimensional, and every irreducible component is isomorphic to the Fermat surface. We also determine its connected components, irreducible components and their intersection behaviors by means of the Bruhat–Tits building of \({{\,\mathrm{PGSp}\,}}_4({\mathbb {Q}}_p)\). In addition, we compute the intersection multiplicity of the GGP cycles associated to an embedding of the considering Rapoport–Zink space into the Rapoport–Zink space for the unramified \({{\,\mathrm{GU}\,}}_{2,2}\) with hyperspecial level for the minuscule case.

Ghost classes in {mathbb {Q}}-rank two orthogonal Shimura varieties

In this article, the existence of ghost classes for the Shimura varieties associated to algebraic groups of orthogonal similitudes of signature (2, n) is investigated. We make use of the study of the weights in the mixed Hodge structures associated to the corresponding cohomology spaces and results on the Eisenstein cohomology of the algebraic group of orthogonal similitudes of signature (1, n-1). For the values of n = 4, 5 we prove the non-existence of ghost classes for most of the irreducible representations (including most of those with an irregular highest weight). For the rest of the cases, we prove strong restrictions on the possible weights in the space of ghost classes and, in particular, we show that they satisfy the weak middle weight property.

Torsion for abelian varieties of type III

Let A be an abelian variety defined over a number field K. The number of torsion points that are rational over a finite extension L is bounded polynomially in terms of the degree [L:K] of L over K. Under the following three conditions, we compute the optimal exponent for this bound in terms of the dimension of abelian subvarieties and their endomorphism rings: (1) A is geometrically isogenous to a product of simple abelian varieties of type I, II or III, according to the Albert classification; (2) A is of “Lefschetz type”, that is, the Mumford–Tate group is the group of symplectic or orthogonal similitudes which commute with the endomorphism ring; (3) A satisfies the Mumford–Tate Conjecture. This result is unconditional for a product of simple abelian varieties of type I, II or III with specific relative dimensions. Further, building on work of Serre, Pink, Banaszak, Gajda and Krasoń, we also prove the Mumford–Tate Conjecture for a few new cases of abelian varieties of Lefschetz type.

Freudenthal triple systems in arbitrary characteristic

Let q:V→K be the form of degree 4 and H the alternating bilinear form that together give rise to a Freudenthal triple system when the characteristic of K is different from 2 and 3. We give a form Θ of degree 4 on a central cover of V over an arbitrary field in arbitrary characteristic and determine the group of similitudes of Θ. This group coincides with the intersection of the group of similitudes of q and the isometry group of H when the characteristic is different from 2 and 3 and has the expected structure of a suitable Levi subgroup in general. We extend known results about the stabilizers of elements of V and orbits of the group of similitudes and show, in particular, the existence of an exotic set of orbits when char(K)=2 and K is not perfect.

Transfer to Characteristic Zero of the Jacquet–Mao’s Metaplectic Fundamental Lemma

Abstract Jacquet conjectured that globally a cuspidal automorphic representation of $\mathrm{G}\mathrm{L}_r$ should be in the image of the metaplectic correspondence precisely when it is distinguished with respect to a split orthogonal similitude group. Jacquet–Mao have suggested an approach to solve this problem by using “relative” traces formula. One of the steps of this approach is the Jacquet–Mao’s metaplectic fundamental lemma. The author proved this fundamental lemma for any $r$ in the case of positive characteristic. The aim of this paper is to extend this result to the more general case.

On the Geometric Side of the Arthur Trace Formula for the Symplectic Group of Rank 2

We study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group or the split symplectic group of rank 2 over any algebraic number field. In particular, we show that the coefficients of unipotent orbital integrals are expressed by the Dedekind zeta function, Hecke L-functions, and the Shintani zeta function for the space of binary quadratic forms.

Transitive groups of similitudes generated by octonions

Abstract We study the structure of various groups generated by (left) multiplications in alternative division algebras, and construct epimorphisms onto orthogonal and special orthogonal groups. Throughout, we include the characteristic two case in our treatment.

Type numbers of quaternion hermitian forms and supersingular abelian varieties

The word \textit{type number} of an algebra means classically the number of isomorphism classes of maximal orders in the algebra, but here we consider quaternion hermitian lattices in a genus and their right orders. Instead of inner isomorphism classes of right orders, we consider isomorphism classes realized by similitudes of the quaternion hermitian forms.The number $T$ of such isomorphism classes are called \textit{type number} or \textit{$G$-type number}, where $G$ is the group of quaternion hermitian similitudes. We express $T$ in terms of traces of some special Hecke operators. This is a generalization of the result announced in [5] (I) from the principal genus to general We also apply our result to the number of isomorphism classes of any polarized superspecial abelian varieties which have a model over ${\Bbb F}_p$ such that the polarizations are in a fixed genus of lattices. This is a generalization of [8] and has an application to the number of components in the supersingular locus which are defined over ${\Bbb F}_p$.

A Factorization Result for Classical and Similitude Groups

AbstractFor most classical and similitude groups, we show that each element can be written as a product of two transformations that preserve or almost preserve the underlying form and whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of Mœglin, Vignéras, and Waldspurger on the existence of automorphisms of p-adic classical groups that take each irreducible smooth representation to its dual.

Faltings heights of abelian varieties with complex multiplication

Let M be the Shimura variety associated with the group of spinor similitudes of a quadratic space over Q of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain L-functions. As an application of this result, we prove an averaged version of Colmez’s conjecture on the Faltings heights of CM abelian varieties.

Clifford parallelisms defined by octonions

We define (left and right) Clifford parallelisms on a seven-dimensional projective space algebraically, using an octonion division algebra. Thus, we generalize the two well-known Clifford parallelisms on a three-dimensional projective space, obtained from a quaternion division algebra. We determine (for both the octonion and quaternion case) the automorphism groups of these parallelisms. A geometric description of the parallel classes is given with the help of a hyperbolic quadric in a Baer superspace, obtained from the split octonion algebra over a quadratic extension of the ground field, again generalizing results that are known for the quaternion case. In contrast to the quaternion case, the orbits of the two Clifford parallelisms under the group of direct similitudes of the norm form of the algebra are non-trivial in the octonion case. The two spaces of parallelisms can be seen as the point sets of two point-line geometries, both isomorphic to the seven-dimensional projective space. Together with the original space, we thus have three versions of this projective space. We introduce a triality between them which is closely related to the triality of the polar space of split type $$\mathrm {D}_4$$ .

On simple polynomial GrT-modules

Using the general framework of polynomial representations defined by Doty and generalizing the definition given by Doty, Nakano and Peters for G=GLn, we consider polynomial representations of GrT for an arbitrary closed reductive subgroup scheme G⊆GLn and a maximal torus T of G in positive characteristic. We give sufficient conditions on G making a classification of simple polynomial GrT-modules similar to the case G=GLn possible and apply this to recover the corresponding result for GLn with a different proof, extending it to symplectic similitude groups, Levi subgroups of GLn and, in a weaker form, to odd orthogonal similitude groups. We also consider orbits of the affine Weyl group and give a condition for equivalence of blocks of polynomial representations for GrT in the case G=GLn.