Related papers: Computing Tangent Spaces to Eigenvarieties

We establish a global weighted $L^p$ estimate for the gradient of the solution to a divergence-form elliptic equations, where the coefficients are in a weighted VMO space and the equations have singularities on a co-dimension two boundary.

In this preprint we prove that any finite slope modular form fits into a p-adic family of modular forms which is indexed by the weight. Here, the term p-adic family means that p-adic congruences between weights entail certain p-adic…

We consider polynomial equations, or systems of polynomial equations, with integer coefficients, modulo prime numbers $p$. We offer an elementary approach based on a counting method. The outcome is a weak form of the Lang-Weil lower bound…

We develop a new strategy for studying low weight specializations of $p$-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate--Vatsal which states that a Hida family contains infinitely…

We compute the volumes of the eigenform loci in the moduli space of genus two Abelian differentials. From this, we obtain asymptotic formulas for counting closed billiards paths in certain L-shaped polygons with barriers.

Here we develop a technique of computing the invariants of $n-$ary forms and systems of forms using the discriminants of corresponding multilinear forms built of their partial derivatives, which should be cosidered as analogues of classical…

The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on…

We compute the waves propagating on a compact 3-manifold of constant positive curvature with a non trivial topology: the Poincar\'e dodecahedral space that is a plausible model of multi-connected universe. We transform the Cauchy problem to…

We study the weight part of Serre's conjecture for generic $n$-dimensional mod $p$ Galois representations. We first generalize Herzig's conjecture to the case where the field is ramified at $p$ and prove the weight elimination direction of…

Let $K$ be an imaginary quadratic field and let $\mathcal{O}_K$ be its ring of integers. For an integral ideal $\mathfrak{n}$ of $\mathcal{O}_K$, let $\Gamma_0({\mathfrak{n}})$ be the congruence subgroup of level ${\mathfrak{n}}$ consisting…

For the weighted Dirac eigenvalue problem, we show that the two-sided weighted spectrum depends continuously on the weight under continuous deformations within a uniformly elliptic class. Moreover, for differentiable families of weights we…

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 investigate the behaviour of the reduction type and related invariants of curves in families of curves over a discretely valued field. By a family, we will mean a set of curves obtained by perturbing the coefficients of the defining…

In the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary…

Let p be an odd prime and g an integer greater or equal to 2. We prove that a finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed over the g-dimensional weight space. The proof of this result relies on the…

We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli…

Starting with a finite-dimensional complex Lie algebra, we extend scalars using suitable commutative topological algebras. We study Birkhoff decompositions for the corresponding loop groups. Some results remain valid for loop groups with…

Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of…

We give an explicit description of the matrix associated to the $U_p$ operator acting on spaces of overconvergent Hilbert modular forms over totally real fields. Using this, we compute slopes for weights in the centre and near the boundary…

We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the…