Related papers: A remark on non-integral p-adic slopes for modular…

This paper gives a general nonsingularity condition for a blowup along an ideal, related to Gaussian curvature.

In this paper, we prove that nonnegative polyharmonic functions on the upper half space satisfying a conformally invariant nonlinear boundary condition have to be the "\emph{polynomials} plus \emph{bubbles}" form. The nonlinear problem is…

Generalising the recent method of Andreatta, Iovita, and Pilloni for cuspidal forms, we construct an eigenvariety for symplectic and unitary groups that parametrises systems of eigenvalues of overconvergent and locally analytic $p$-adic…

We give a sufficient condition to rule out complete Riemannian metrics with nonnegative scalar curvature on the interiors of handlebodies. In higher dimensions, we give examples of ends of manifolds with positive scalar curvature metrics.

We show that a quartic $p$-adic form with at least $3192$ variables possesses a non-trivial zero. We also prove new results on systems of cubic, quadratic and linear forms. As an example, we show that for a system comprising two cubic forms…

We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing "invariant versions" of iterated integrals of modular forms. The construction will be based on an extension of…

In this paper, we consider modular forms for finite index subgroups of the modular group whose Fourier coefficients are algebraic. It is well-known that the Fourier coefficients of any holomorphic modular form for a congruence subgroup…

For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced…

This article extends our study of the geometry of the $p$-adic eigencurve at a point defined by a weight $1$ cuspform $f$ irregular at $p$ and having complex multiplication, and the implications in Iwasawa and in Hida theories. The novel…

We provide an integral estimate for a non-divergence (non-variational) form second order elliptic equation $a_{ij}u_{ij}=u^p$, $u\ge 0$, $p\in[0, 1)$, with bounded discontinuous coefficients $a_{ij}$ having small BMO norm. We consider the…

We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincar\'e series in a companion paper. The source term of the Laplace equation is a product of…

To have an uniform estimate for the solutions of the scalar curvature equation perturbed by a non linear term, we give some minimal condition on the scalar curvature.

We derive slow-roll conditions for a scalar field which is non-minimally coupled with gravity in a consistent manner and express spectral indices of scalar/tensor perturbations in terms of the slow-roll parameters. The conformal invariance…

In this paper, we give some effective bounds for the $j$-invariant of integral points on arbitrary modular curves over arbitrary number fields assuming that the number of cusps is not less than 3.

Several authors have recently proved results which express cusp forms as $p$-adic limits of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. The proofs involve techniques from the theory of weak harmonic…

We show that all $p$-adic quintic forms in at least $n>4562911$ variables have a non-trivial zero. We also derive new result concerning systems of cubic and quadratic forms.

We construct level-raising congruences between $p$-ordinary automorphic representations, and apply this to the problem of symmetric power functoriality for Hilbert modular forms. In particular, we prove the existence of the $n^\text{th}$…

This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These…

By adopting a new approach to the analysis of the density of p-adic solutions arising in applications of the circle method, we show that under modest conditions the existence of non-trivial p-adic solutions suffices to establish positivity…

We prove level raising results for $p$-adic automorphic forms on definite unitary groups $U(3)/\mathbb{Q}$ and deduce some intersection points on the eigenvariety. Let $l$ be an inert prime where the level subgroups varies, if there is a…