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It is known that the L-function of an elliptic curve defined over Q is given by the Mellin transform of a modular form of weight 2. Does that modular form have anything to do with string theory? In this article, we address a question along…
We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be defined in any finite-dimensional Hilbert space. It is defined by using the Kravchuk polynomials, and we call it…
We prove a noncompact Serre-Swan theorem characterising modules which are sections of vector bundles not necessarily trivial at infinity. We then identify the endomorphism algebras of the resulting modules. The endomorphism results continue…
We give a classification of the simple modules for the cyclotomic Hecke algebras over $\mathbb{C}$ in the modular case. We use the unitriangular shape of the decomposition matrices of Ariki-Koike algebras and Clifford theory.
Let $G$ be a connected reductive group over a totally real field $F$ which is compact modulo center at archimedean places. We find congruences modulo an arbitrary power of p between the space of arbitrary automorphic forms on $G(\mathbb…
We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals with elliptic modular forms and their derivatives, and generalizes the Rankin-Cohen…
We study the unitary representations of the non-compact real forms of the complex Lie superalgebra sl(n|m). Among them, only the real form su(p,q|m) (p+q=n) admits nontrivial unitary representations, and all such representations are of the…
We introduce a method that is based on Fourier series expansions related to Jacobi elliptic functions and that we apply to determine new identities for evaluating hyperbolic infinite sums in terms of the complete elliptic integrals $K$ and…
We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients…
We study the cohomology of Lie superalgebras for the full complex of forms: superforms, pseudoforms and integral forms. We use the technique of spectral sequences to abstractly compute the Chevalley-Eilenberg cohomology. We first focus on…
We construct explicit Eichler-Shimura morphisms for families of overconvergent Siegel modular forms of genus two. These can be viewed as $p$-adic interpolations of the Eichler-Shimura decomposition of Faltings-Chai for classical Siegel…
In this paper, by modifying significantly the Friedrichs-Gross mollifier technique and/or using the Lasry-Lions regularization technique together with some carefully chosen cut-off functions, for the first time we construct explicitly…
Recent work of Altu\u{g} continues the preliminary analysis of Langlands' Beyond Endoscopy proposal for $GL(2)$ by removing the contribution of the trivial representation to the trace formula using a Poisson summation formula. We show that…
We study the Selmer group associated to a $p$-ordinary newform $f \in S_{2r}(\Gamma_0(N))$ over the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K/\mathbb{Q}$. Under certain assumptions, we prove that this Selmer…
The paper discusses relations between the structure of the complex Fermi surface below the spectrum of a second order periodic elliptic equation and integral representations of certain classes of its solutions. These integral…
We construct a family of integrable vertex model based on the typical four-dimensional representations of the quantum group deformation of the Lie superalgebra $sl(2|1)$. Upon alternation of such a representation with its dual this model…
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_\alpha$ were introduced…
It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $\hat{\frak{g}}$ span an $SL_2(\mathbf{Z})$-invariant space. This result extends to admissible…
We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds ${\cal M}=$SL$(2,\mathbb R)$ and ${\cal M}=$ SL$(2,\mathbb R)/U(1)$ to a finite-dimensional simple Lie group $G$. This construction is…
We classify the possible finite symmetries of conformal field theories with an affine Lie algebra su(2) and su(3), and discuss the results from the perspective of the graphs associated with the modular invariants. The highlights of the…