In this paper we deal with monogenic and $k$-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between $k$-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vanishing monogenic automorphic form a Clifford algebra valued Maaß wave form.
In the second part of the paper we compute the Fourier expansion of the $k$-hypermonogenic Eisenstein series which provide us with the simplest non-vanishing examples of $k$-hypermonogenic automorphic forms.
In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator $D Δ^{k/2}$ for some even $k \in {\mathbb{Z}}$. They will be called $k$-holomorphic Cliffordian automorphic forms. $k$-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions have also the property that they are solutions to the homogeneous and inhomogeneous Weinstein equation. This function class includes the set of $k$-hyp…
This paper deals with a new analytic type of vector- and Clifford algebra valued automorphic forms in one and two vector variables. For hypercomplex generalizations of the classical modular group and their arithmetic congruence subgroups Eisenstein- and Poincaré type series that are annihilated by Dirac operators, and more generally, by iterated Dirac operators on the upper half-space of $R^n$ are discussed. In particular we introduce (poly-)monogenic modular forms on hypercomplex generalizations of the classical theta group.
We characterize the space of the so-called planar mixed automorphic forms of type $(ν,μ)$ with respect to an equivariant pair $(ρ,τ)$ as the image, by an appropriate transform, of the usual (Landau) automorphic forms involving special multiplier.
In this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily $K_\infty$-finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a ''smooth version'' of the space of automorphic forms, whose internal structure was the topic of a famous paper of Franke. We prove that the important decomposition along the parabolic support, and the even finer - and structurally more important - decomposition along the cuspidal su…
We show the existence of some non-classical cohomological p-adic automorphic eigenforms for SL(2) using endoscopy and the geometry of eigenvarieties. These forms seem to account for some non-automorphic members of classical global L-packets.
We prove the subconvexity conjecture for sup-norms of automorphic forms for congruence subgroups of SL(n, Z) that satisfy the Ramanujan conjecture at infinity.
Higher order automorphic forms have recently been introduced to study important questions in number theory and mathematical physics. We investigate the connection between these functions and Chen's iterated integrals. Then using Chen's theory, we prove a structure theorem for automorphic forms of all orders. This allows us to define an analogue of a mixed Hodge structure on a space of higher order automorphic forms.
Second-order automorphic forms are similar to the usual automorphic forms but have a weaker automorphy condition. We answer a question of Zagier and find the dimensions of spaces of holomorphic, even weight, second-order forms. We also establish a cohomological interpretation.
We construct extensions of the field of rational numbers with the Galois group G_2(F_p) by reducing p-adic representations attached to automorphic representations.
We study linear ordinary differential equations which are analytically parametrized on Hermitian symmetric spaces and invariant under the action of symplectic groups. They are generalizations of the classical Lamé equation. Our main result gives a closed relation between such differential equations and automorphic forms for symplectic groups. Our study is based on techniques concerning with the monodromy of complex differential equations, the Baker-Akhiezer functions and algebraic curves attached to rings of differential operators.
We consider certain Dirichlet series of Selberg type, constructed from periods of automorphic forms. We study analytic properties of these Dirichlet series and show that they have analytic continuation to the whole complex plane.
We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If C denotes the congruence module for a fixed automorphic Hecke eigenform π_0 we prove an exact relation between the p-adic valuation of the order of C and the sum of the exponents of p-power congruences between the Hecke eigenvalues of π_0 and other automorphic forms. We apply this result to several situations including the congruences described by Mazur's Eisenstein ideal.
We show that there are only finitely many nonconstant reflective automorphic forms $Ψ$ on even lattices of squarefree level splitting two hyperbolic planes and give a complete classification in the case where the zeros of $Ψ$ are simple and $Ψ$ has singular weight.
We give an asymptotic formula for the number of automorphic forms on the non-split norm one torus $T$ associated with an imaginary quadratic extension of $\mathbb{Q}$, ordered by analytic conductor.
We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem for family of primitive cusp forms of weight $2$ and prime level, and discuss under which conditions the argument will apply to general reasonable family of automorphic $L$-functions.
We construct Eigenvarieties for PEL Shimura varieties which interpolate cuspidal, finite slope automorphic forms for PEL Shimura varieties appearing as global sections of (coherent) automorphic sheaves, under the hypothesis that the primes are unramified in the Shimura datum, but allowing the ordinary locus to be empty. We then use this construction to extend previous results on the Bloch-Kato conjecture for Hecke characters of a quadratic imaginary field.
In this paper, we give a geometric construction of the Jacquet-Langlands transfer for automorphic forms of higher weights by studying the geometry of the mod $p$ fibres of different Hodge type Shimura varieties which satisfy a mild assumption and the cohomological correspondences between them.
The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. In the first place, we combine the Ichino--Ikeda--Neal-Harris (IINH) formula -- which is now a theorem -- with an analysis of cup products of coherent cohomological automorphic forms on Shimura varieties to establish relations between certain automorphic periods and critical values of Rankin-Selberg and Asai $L$-functions of ${\rm GL}(n)\times{\rm GL}(m)$ over CM …
This is an expanded set of notes based on two lectures given by the author at the 2022 IHES summer school on the Langlands program, on explicit techniques for the construction of automorphic representations.
We prove the optimal upper bound sum || f ||_4^4 << q^{eps} where f runs over an orthonormal basis of Maass cusp forms of prime level q and bounded spectral parameter.
We consider the genus of $20$ classes of unimodular Hermitian lattices of rank $12$ over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global Arthur parameters for the associated automorphic representations, which recover the computed Hecke eigenvalues. Congruences between Hecke eigenspaces, combined with the assumed parameters, recover known congruences for classical modular forms, and support new instances of conjectured Eisenstein congruences for $\mathbb{U}(2,2)$ automorphic forms.
In this article, we establish polynomial-growth bound for the sequence of Fourier coefficients associated to even integer weight vector-valued automorphic forms of Fuchsian groups of the first kind. At the end, their $L$-functions and exponential sums have been discussed.
We show that the systems of prime-to-$p$ Hecke eigenvalues arising from automorphic forms$\pmod p$ for a good prime $p$ associated to an algebraic group $G/\mathbb Q$ of Hodge type are the same as those arising from algebraic modular forms$\pmod p$ associated to an inner form of $G$.
We prove various results on the cohomology of arithmetic lattices arising from quaternion algebras over a number field with at least one complex place, including a strong restriction on the allowable weights of cuspidal cohomological automorphic forms and a conditional power saving for their multiplicities when the associated locally symmetric space is a hyperbolic 3-manifold.
We construct a p-adic Eisenstein measure with values in the space of vector-weight p-adic automorphic forms on certain unitary groups. This measure allows us to p-adically interpolate special values of certain vector-weight C-infinity automorphic forms, including Eisenstein series, as their weights vary. We also explain how to extend our methods to the case of Siegel modular forms and how to recover Nicholas Katz's p-adic families of Eisenstein series for Hilbert modular forms.
Let $\mathbb{A}$ be the adele ring of a totally real algebraic number field $F$. We push forward an explicit computation of a relative trace formula for periods of automorphic forms along a split torus in $GL(2)$ from a square free level case done by Masao Tsuzuki, to an arbitrary level case. By using a relative trace formula, we study central values of automorphic $L$-functions for cuspidal automorphic representations of $GL(2, \mathbb{A})$ corresponding to Maass forms with arbitrary level.
In this article, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [3] and [9] to higher dimensions.
Let q be an integral quadratic form of signature (2,m+2). We will show that the Siegel theta functions attached to q satisfies certain symmetries. As an application, we prove the symmetries for automorphic forms on the orthogonal group of q closely related to Heegener divisors (Borcherds lifts and automorphic Green functions).
In this paper we explore some properties of periods attached to automorphic representations of unitary groups over CM fields and the critical values of their $L$-functions. We prove a formula expressing the critical values in the range of absolute convergence in terms of Petersson norms of holomorphic automorphic forms. On the other hand, we express the Deligne period of a related motive as a product of quadratic periods and compare the two expressions by means of Deligne's conjecture.
I present a general theory of overconvergent p-adic automorphic forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions due to Buzzard, Chenevier and Yamagami for forms of GL(n). This leads to some new phenomena, including the appearance of intermediate spaces of "semi-classical" automorphic forms; this gives a hierarchy of interpolation spaces (eigenvarieties) interpolating classical automorphic forms satisfying different finite slope conditions (corresponding to a choice of parabolic subgroup of G at p).…
A general Voronoï summation formula for the (metaplectic) double cover of $\text{GL}_2$ is derived via the representation theoretic framework à la Ichino--Templier. The identity is also formulated classically and used to establish Voronoï summation formulae for half-integral weight modular forms and Maaß forms.
We present an approach to proving control theorems for overconvergent automorphic forms on some Harris-Taylor unitary Shimura varieties based on a comparison between the rigid coho- mology of the multiplicative ordinary locus and the rigid cohomology of the overlying Igusa tower, the latter which may be computed using the Harris-Taylor version of the Langlands-Kottwitz method. We also prove a higher level version, generalizing work of Coleman.
We define two-parameter generalizations of Andrews' $(k+1)$-marked odd Durfee symbols and $2k$th symmetrized odd rank moments, and study the automorphic properties of some of their generating functions. When $k=0$ we obtain families of modular forms and mock modular forms. When $k \geq 1$, we find quasimodular forms and quasimock modular forms.
We prove an unconditional power saving for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on GL_2 over any number field which is not totally real. Our proof involves the theory of p-adically completed cohomology developed by Calegari and Emerton, and a bound for the growth of coinvariants in certain finitely generated non-commutative Iwasawa modules.
In this paper we generalize the work of Harris-Soudry-Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on GL_2 over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semisimplification.
I give an algorithm for computing the full space of automorphic forms for definite unitary groups over Q, and apply this to calculate the automorphic forms of level $G(Z-hat)$ and various small weights for an example of a rank 3 unitary group. This leads to some examples of various types of endoscopic lifting from automorphic forms for U_1 x U_1 x U_1 and U_1 x U_2, and to an example of a non-endoscopic form of weight (3,3) corresponding to a family of 3-dimensional irreducible l-adic Galois representations. I also compute the 2-adic slopes of some automorphic forms with level structure at 2, …
We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky--Nelson, corresponding to multiplicative Dirichlet characters, and applies in particular to trace functions of small conductor modulo primes.
We investigate the relations for $L$-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and half-integral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of $SL(2)$. Our goal is converse theorems for automorphic distributions and Maass forms of level $N$ characterizing them by analytic properties of the associated $L$-functions. As an application of our converse theorems, we construc…
We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. Knopp generalized this to real weights. We show that for weights that are not an integer at least 2 the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. We impose no condition on the growth of the automorphic forms at the cusps. For real weights that are not an int…
We prove for L-function attached to an automorphic cusp form for the Hecke congruence group $Γ_0(D)$, which is also an eigenfunction of all the Hecke operators, that a positive proportion of its non-trivial zeros lie on the critical line. This result extends the work of J.L. Hafner of 1983 where the case of the full modular group is considered.
We define weight changing operators for automorphic forms on Grassmannians, i.e., on orthogonal groups, and investigate their basic properties. We then evaluate their action on theta kernels, and prove that theta lifts of modular forms, in which the theta kernel involves polynomials of a special type, have some interesting differential properties.
We present a level raising result for families of p-adic automorphic forms for a definite quaternion algebra D over the rational numbers. The main theorem is an analogue of a theorem for classical automorphic forms due to Diamond and Taylor. We show that certain families of forms old at a prime l intersect with families of l-new forms (at a non-classical point). One of the ingredients in the proof of Diamond and Taylor's theorem (which also played a role in earlier work of Taylor) is the definition of a suitable pairing on the space of automorphic forms. In our situation one cannot define such…
We prove an integrality result for the value at s=1 of the adjoint L-function associated to a cohomological cuspidal automorphic representation on GL(n) over any number field. We then show that primes (outside an exceptional set) dividing this special value give rise to congruences between automorphic forms. We also prove a non-vanishing property at infinity for the relevant Rankin-Selberg L-functions on GL(n) x GL(n).
It is known due to Baker and Montgomery that almost all Fekete polynomials under certain ordering have at least one zero on the interval (0, 1). In terms of the positive-definiteness, Fekete polynomial has no zero on the interval (0, 1) if and only if the corresponding automorphic form is positive-definite. On generalizing their result, we formulate an axiomatic result about sets of automorphic forms satisfying certain averages when suitably ordered, which ensures that almost all forms are not positive-definite within such sets. We then apply the result to various families, including the famil…
The zeros and poles of standard automorphic $L$-functions attached to representations of classical groups are linked to the nonvanishing of lifts in the theory of the theta correspondence. The results of this paper show that when a cuspidal representation $σ$ of a symplectic group $G$ lifts to a cuspidal representation $π= θ_ψ(σ)$ of an orthogonal group $H$ attached to a quadratic space $V$ of dimension $m$, the Fourier coefficients of automorphic forms in $σ$ are linked to periods of automorphic forms in $π$. Consequently, when our results are combined with the Rallis inner product formula in…
The goal of this paper is to study certain p-adic differential operators on automorphic forms on U(n,n). These operators are a generalization to the higher-dimensional, vector-valued situation of the p-adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the p-adic case of the C^{\infty}-differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain p-adic L-functions attached to p-adic families of automorphic forms on the unitary grou…
In this paper a theory of Hecke operators for higher order modular forms is established. The definition of cusp forms and attached L-functions is extended beyond the realm of parabolic invariants. The role of representation theoretic methods is clarified and, motivated by higher order forms, new convolution products of L-functions are introduced.
We extend some results of Gun, Murty, and Rath on elliptic modular forms. We take ANY Fuchsian triangle group with a cusp and look at power series expansions in a natural parameter around that cusp. Consider the automorphic forms for such a triangle group whose power series expansions in the natural parameter have algebraic coefficients. We show that the zeros of such forms are either transcendental or are "CM." By "CM," we mean they correspond to abelian varieties with complex multiplication. This result is the first of its kind in the case of non-arithmetic groups.
