arXiv subjects
Publications and source records attributed to Mehmet Haluk Sengun.
This is a survey of the recent works that describe theta correspondence as a C*-correspondence. The main mechanism is elucidated in the special case when one of the groups involved in the theta correspondence is compact.
In analogy with the construction of representations of adelic groups as restricted products of representations of local groups, we study restricted tensor products of Hilbert C*-modules and of C*-correspondences. The construction produces global C*-correspondences from compatible collections of local C*-correspondences. When applied to the collection of C*-correspondences capturing local parabolic induction, the construction produces a global C*-correspondence that captures adelic parabolic induction.
We prove that the well-known explicit construction of the local theta correspondence by Li has a simple interpretation in terms of group C*-algebras. In particular, we deduce that in two standard cases where Li's method work, local theta correspondence arises from a continuous functor. Moreover, using results from a companion paper, we treat global theta correspondence using C*-algebraic methods. As a byproduct, we exhibit that Rallis inner product formula can be interpreted as a certain natural inclusion being an isometry.
Let (G,H) be one of the equal rank reductive dual pairs (Mp_{2n},O_{2n+1}) or (U_n,U_n) over a non-archimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain subsets, say R(G) and R(H), of the tempered duals of G and H. We prove that this bijection arises from an equivalence between the categories of representations of two C*-algebras whose spectra are R(G) and R(H). This equivalence is implemented by the induction functor associated to a Morita equivalence bimodule (in the sense of Rieffel) which we construct using the oscillator representation. As an immediate corollary, we deduce that the bijection is functorial and continuous with respect to weak inclusion. We derive further consequences regarding the transfer of characters and preservation of formal degrees.
Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L^2(G/H) associated to test functions. In this paper we present a cohomological interpretation of the trace formula involving the K-theory of the maximal group C*-algebras of G and H. As an application, we exploit the role of group C*-algebras as recipients of higher indices of elliptic differential operators and we obtain the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici from ours.
We discuss Base Change functoriality for mod p eigenforms for GL(2) over number fields. We carry out systematic computer experiments and collect data supporting its existence in cases of field extensions K/F where F is imaginary quadratic and K is CM quartic.
Let $Γ$ be a lattice in a locally compact group $G$. In earlier work, we used $KK$-theory to equip the $K$-groups of any $Γ$-$C^{*}$-algebra on which the commensurator of $Γ$ acts with Hecke operators. When $Γ$ is arithmetic, this gives Hecke operators on the $K$-theory of certain $C^{*}$-algebras that are naturally associated with $Γ$. In this paper, we first study the topological $K$-theory of the arithmetic manifold associated to $Γ$. We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Kasparov product and thus that the $KK$-groups associated to an arithmetic group $Γ$ become true Hecke modules. We conclude by discussing Hecke equivariant maps in $KK$-theory in great generality and apply this to the Borel-Serre compactification as well as various noncommutative compactifications associated with $Γ$. Along the way we discuss the relation between the $K$-theory and the integral cohomology of low-dimensional manifolds as Hecke modules.
Let $Γ$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $Γ$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in KK-theory, we endow the K-groups of the arithmetic manifold $X/Γ$, of the reduced group C*-algebra of $Γ$ and of the boundary crossed product algebra associated to the action of $Γ$ on $\partial X$, with Hecke operators. The K-theory and K-homology groups of these C*-algebras are related by a Gysin six-term exact sequence. In the case when $Γ$ is a group of real hyperbolic isometries, we show that this Gysin sequence is Hecke equivariant. Finally, in the case when $Γ$ is a subgroup of a Bianchi group, we construct explicit Hecke-equivariant maps between the integral cohomology of $Γ$ and each of these K-groups. Our methods apply to torsion-free finite index subgroups of $PSL(2,\mathbb{Z})$ as well. These results are achieved in the context of unbounded Fredholm modules, shedding light on noncommutative geometric aspects of the purely infinite boundary crossed product algebra.
Assuming two deep but standard conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields Q(\sqrt{-d}) with -d=2, 3 mod 4. For a general number field K, again assuming standard conjectures, we give a criterion based on the solutions to a certain S-unit equation, which if satisfied implies the asymptotic Fermat's Last Theorem.
The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve E_f over K. In our previous paper, we associated, building on works of Darmon and Greenberg, a p-adic lattice to f, under certain hypothesis, and implicitly conjectured that this lattice is commensurable with the p-adic Tate lattice of E_f . In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from f, an explicit Weierstrass equation for the conjectural E_f . We develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we compute Weierstrass equations of hundreds of elliptic curves, some with huge heights, over dozens of number fields. The data we obtain provide overwhelming amount of support for the conjecture and furthermore demonstrate that the conjecture provides an efficient tool to building databases of elliptic curves over number fields.
We present new constructions of complex and p-adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture.
We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. For this we use archimedean results from Harris, Soudry, Taylor and replace the global arguments of Roberts by the non-vanishing result of Takeda. As an application of our lifting result, we exhibit an abelian surface $B$ defined over $\mathbb{Q}$, which is not restriction of scalars of an elliptic curve and satisfies the Brumer-Kramer Paramodularity Conjecture.
These are the notes of the three lectures I delivered at the mini-workshop "Knot Theory and Number Theory around the A-Polynomial" at the Instituto Superior Tecnico (IST) in Lisbon in January 2014. The goal of the lectures was to familiarize, both the author and, the audience with the A-polynomials and the connection between the Mahler measures of A-polynomials and volumes. The style of these notes is expository, written informally with the aim of giving a flavor of the subject with ample amount of references to direct the interest readers to the details.
Let M be an arithmetic hyperbolic 3-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of M, where each basis element is represented by a surface of `low' genus, and give evidence for this. We explain the relationship between this conjecture and the study of torsion homology growth.
Using Lefschetz numbers of certain involutions, we provide lower bounds for the cuspidal cohomology of principal congruence subgroups of Bianchi groups. The asymptotic lower bounds that follow from our results complement recent results of Calegari-Emerton, Marshall and Finis-Grunewald-Tirao. Moreover, we discuss the relationship between these involutions and the base change classes in the cohomology.
In this note, we work out the dimension of the subspace of base-change forms and their twists inside the space of Bianchi modular forms with fixed (Galois stable) level and weight.
These are the extended notes of a talk I gave at the Geometric Topology Seminar of the Max Planck Institute for Mathematics in Bonn on January 30th, 2012. My goal was to familiarize the topologists with the basics of arithmetic hyperbolic 3-manifolds and sketch some interesting results in the theory of 3-manifolds (such as Labesse-Schwermer, Calegari-Dunfield, Dunfield-Ramakrishnan) that are obtained by exploiting the connections with number theory and automorphic forms. The overall intention was to stimulate interaction between the number theorists and the topologists present at the Institute.
For a quadratic field K, we investigate continuous mod p representations of the absolute Galois groups of K that are unramified away from p and infinity. We prove that for certain pairs (K,p), there are no such irreducible representations. We also list some imaginary quadratic fields for which such irreducible representations exist. As an application, we look at elliptic curves with good reduction away from 2 over quadratic fields.