Related papers: Combinatorial Congruences and $\psi$-Operators

We study the behavior of the Iwasawa invariants of the Iwasawa modules which appear in Kato's main conjecture without $p$-adic $L$-functions under congruences. It generalizes the work of Greenberg-Vatsal, Emerton-Pollack-Weston, B.D. Kim,…

We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as…

In this paper we study effective recursion formulae for computing intersection numbers of mixed $\psi$ and $\kappa$ classes on moduli spaces of curves. By using the celebrated Witten-Kontsevich theorem, we generalize Mulase-Safnuk form of…

In the present paper, we study regularity of the Andersson-Samuelsson Koppelman integral operator on affine cones over smooth projective complete intersections. Particularly, we prove $L^p$- and $C^\alpha$-estimates, and compactness of the…

We deduce the cyclotomic Iwasawa main conjecture for Hilbert modular cuspforms with complex multiplication from the multivariable main conjecture for CM number fields. To this end, we study in detail the behaviour of the $p$-adic…

The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L_p^\sharp(f,T) and L_p^\flat(f,T) for a weight two modular form \sum a_n q^n and a good prime p. This generalizes work of Pollack who…

In [Tha15], we looked at two (`multiplicative' and `Carlitz-Drinfeld additive') analogs each, for the well-known basic congruences of Fermat and Wilson, in the case of polynomials over finite fields. When we look at them modulo higher…

We prove Wise's $W$-cycles conjecture. Consider a compact graph $\Gamma'$ immering into another graph $\Gamma$. For any immersed cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$ from the circular components $\mathbb{S}$ of the…

In this work we prove the so-called "torsion congruences" between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative…

We construct a new class of Iwasawa modules, which are the number field analogues of the p-adic realizations of the Picard 1-motives constructed by Deligne in the 1970s and studied extensively from a Galois module structure point of view in…

We give the geometric version of a construction of Colmez-Niziol which establishes a comparison theorem between arithmetic p-adic nearby cycles and syntomic sheaves. The local construction of the period isomorphism uses…

The cyclic sieving phenomenon provides a link between a polynomial analogue of Gauss congruence known as $q$-Gauss congruence, and a combinatorial analogue of Gauss congruence based on sequences of cyclic group actions. We strengthen this…

We use homotopy theoretic methods to prove congruence relations of number theoretic interest. Specifically, we use the theory of $\mathbb E_\infty$ complex orientations to establish $p$-adic K\"ummer congruences among iterated derivatives…

The notion of strong Frobenius structure is classically studied in the theory of $p$-adic differential operators. In the present work, we introduce a new definition of the notion of strong Frobenius structure for $q$-difference operators.…

We prove congruence relations modulo cyclotomic polynomials for multisums of $q$-factorial ratios, therefore generalizing many well-known $p$-Lucas congruences. Such congruences connect various classical generating series to their…

Norm-compatible families of cohomology classes for Shimura varieties, and other arithmetic symmetric spaces, play an important role in Iwasawa theory of automorphic forms. The aim of this note is to give a systematic approach to proving…

Let p be any prime and a be a positive integer. For nonnegative integers l,n and an integer r, the normalized cyclotomic $\psi$-coefficient $${n,r}_{l,p^a}:=p^{-[(n-p^{a-1}-lp^a)/(p^{a-1}(p-1))]} \sum_{k=r(mod p^a)}(-1)^k{n \choose…

In this paper, we construct the cut-and-join operator description for the generating functions of all intersection numbers of $\psi$, $\kappa$, and $\Theta$ classes on the moduli spaces $\overline{\mathcal M}_{g,n}$. The cut-and-join…

The Composite Operator Method (COM) is formulated, its internals illustrated in detail and some of its most successful applications reported. COM endorses the emergence, in strongly correlated systems (SCS), of composite operators,…

We characterize the convex-cyclic weighted composition operators $W_{(u,\psi)}$ and their adjoints on the Fock space in terms of the derivative powers of $ \psi$ and the location of the eigenvalues of the operators on the complex plane.…