Extensions of representations of p-adic groups
We calculate extensions between certain irreducible admissible representations of p-adic groups.
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arXiv subjects
Dipendra Prasad
Publications and source records attributed to Dipendra Prasad.
A local global question in automorphic forms
In this paper, we consider the $\SL(2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\GL(2)$: one due to Harder-Langlands-Rapoport, and the other due to Waldspurger.
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Restrictions of representations of classical groups: examples
In an earlier paper, we considered several restriction problems in the representation theory of classical groups over local and global fields. Assuming the Langlands-Vogan parameterization of irreducible representations, we formulated precise conjectures for the solutions of these restriction problems. In the local case, our conjectural answer is given in terms of Langlands parameters and certain natural symplectic root numbers associated to them. In the global case, the conjectural answer is expressed in terms of the central critical value or derivative of a global $L$-function. In this paper we verify several of these conjectures in certain low rank cases, using methods of base change, theta correspondence, and global arguments.
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Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
We consider several questions about restriction of representations of classical and metaplectic groups over local and global fields to subgroups, extending considerably the scope of the earlier work on $SO(n),SO(n-1)$. This includes Bessel and Fourier-Jacobi models too. We formulate several conjectures about these restriction problems involving root numbers of symplectic representations in the local case, and central critical L-value in the global case. Along the way we prove several results both in number theory and representation theory.
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Selfdual representations of division algebras and Weil groups: A contrast
Selfdual representations of any group fall into two classes when they are irreducible: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the irreducible representations σof the Weil group of a local field k of dimension n with the irreducible representations πof the invertible elements of a division algebra D over k of index n, takes selfdual representations to selfdual representations. In this paper we use global methods to study how the Langlands correspondence behaves relative to this distinction among selfdual representations. We prove in particular that for n even, σis symplectic if and only if πis orthogonal. Our results treat more generally the case of GL_m(B), for B a division algebra over k of index r, and n=mr.
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Bessel models for $GSp(4)$
Methods of theta correspondence are used to analyze local and global Bessel models for $GSp(4)$ proving a conjecture of Gross and Prasad which describes these models in terms of local epsilon factors in the local case, and the non-vanishing of central critical $L$-value in the global case.
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Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups
In this paper we show how a theorem of Jantzen relating the character of a finite dimensional irreducible representation of a disconnected semisimple algebraic group on elements outside the connected component of identity to character of an irreducible representation of another connected semisimple algebraic group can be deduced from properties of the canonical bases.
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Generalised form of a conjecture of Jacquet and a local consequence
Following the work of Harris and Kudla we prove a more general form of a conjecture of Jacquet relating the non-vanishing of a certain period integral to non-vanishing of the central critical value of a certain $L$-function. As a consequence we deduce certain local results about the existence of $GL_2(k)$-invariant linear forms on irreducible, admissible representations of $GL_2({\Bbb K})$ for ${\Bbb K}$ a commutative semi-simple cubic algebra over a non-archimedean local field $k$ in terms of certain local epsilon factors which were proved only in certain cases by the first author in his earlier work. This has been achieved by globalising a locally distinguished representation to a globally distinguished representation, a result of independent interest.
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On the SL(2) period integral
Let E/F be a quadratic extension of number fields. For a cuspidal representation $π$ of SL(2,A_E), we study the non-vanishing of the period integral on SL(2,F)\SL(2,A_F). We characterise the non-vanishing of the period integral of $π$ in terms of $π$ being generic with respect to characters of E\A_E which are trivial on A_F. We show that the period integral in general is not a product of local invariant functionals, and find a necessary and sufficient condition when it is. We exhibit cuspidal representations of SL(2,A_E) whose period integral vanishes identically while each local constituent admits an SL(2)-invariant linear functional. Finally, we construct an automorphic representation $π$ on SL(2,A_E) which is abstractly SL(2,A_F) distinguished but none of the elements in the global L-packet determined by $π$ is distinguished by SL(2,A_F).
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Relating invariant linear form and local epsilon factors via global methods
{We use the recent proof of Jacquet's conjecture due to Harris and Kudla, and the Burger-Sarnak principle to give a proof about the relationship between the existence of trilinear forms on representations of $GL_2(k_u)$ for a non-Archimedean local field $k_u$ and local epsilon factors which was earlier proved only in the odd residue characteristic by this author. The same method gives a global proof of a theorem of Saito and Tunnell about characters of $GL_2$ using a theorem of Waldspurger about period integrals for $GL_2$ and also an extension of the theorem of Saito-Tunnell by this author which was earlier proved only in odd residue characteristic.}
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On certain multiplicity one theorems
We prove several multiplicity one theorems in this paper. For k a local field not of characteristic two, and V a symplectic space over k, any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the symplectic group Sp(V). We prove the analogous result for GO(V) and O(V), where V is an orthogonal space over k. When k is non-archimedean, we prove the uniqueness of Fourier-Jacobi models for representations of GSp(4), and the existence of such models for supercuspidal representations of GSp(4).
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Reduction of homomorphisms mod p and algebraicity
{Let $K$ be a number field, and $A_1,A_2$ abelian varieties over $K$. Let $P$ (resp. $Q$) be a non-torsion point in $ A_1(K)$ (resp. $A_2(K)$) such that for almost all places $v$ of $K$, the order of $Q$ mod $v$ divides the order of $P$ mod $v$. Then we prove (under some conditions on $A_i, i=1,2$) that there is a homomorphism $j$ from $A_1$ to $A_2$ such that $j(P) = Q$. We formulate and extend such a result for any subgroups of $A_i(K), i=1,2$. These results in particular extend the work of Corrales and Schoof from elliptic curves to abelian varieties.
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An analogue of a conjecture of Mazur a question in Diophantine approximation on tori
B. Mazur has considered the question of density in the Euclidean topology of the set of ${\Bbb Q}$-rational points on a variety $X$ defined over ${\Bbb Q}$, in particular for Abelian varieties. In this paper we consider the question of closures of the image of finitely generated subgroups of $T({\Bbb Q})$ in $Γ\backslash T({\Bbb R})$ where $T$ is a torus defined over ${\Bbb Q}$, $Γ$ an arithmetic subgroup such that $Γ\backslash T({\Bbb R})$ is compact. Assuming Schanuel's conjecture, we prove that the closures correspond to sub {\it algebraic} tori of $T$.
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On an Archimedean analogue of Tate's conjecture
We consider an Archimedean analogue of Tate's conjecture, and verify the conjecture in the examples of isospectral Riemann surfaces constructed by Vigneras and Sunada. We also enunciate a simple lemma in group theory which lies at the heart of T. Sunada's theorem about isospectral manifolds.
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Reduction of abstract homomorphisms of lattices mod p and rigidity
In this paper we pose and answer the following question in a few different contexts: Given a homomorphism f:L_1 --> L_2 of a ``lattices'' that ``reduces mod p'' for almost all primes p, is f ``algebraic''? For instance the lattices may be the Mordell-Weil lattices of rational points of abelian varieties over Q, or arithmetic groups etc. Implicit in an affirmative answer to the question for Mordell-Weil lattices is a novel criterion for abelian varities to be isogenous.
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Bounding the torsion in CM elliptic curves
Merel has shown that the order of torsion subgroup of an elliptic curve over a number field can be bounded in terms of only the degree of the number field. The purpose of this note is to investigate what could be the `right bound'. In this paper we use the result of Deuring on the supersingular primes for a CM elliptic curves to give estimate on the number of torsion points on CM elliptic curves over any number field which depends only linearly with the degree of the number field. This result is also a simple corollary of a result of A. Silverberg bounding the order of torsion elements which was proved using the main theorem of Complex multiplication. We also give a heuristic for hoping that the number of torsion points on any elliptic curve be a bounded linearly with the degree of the number field.
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