相关论文: A counterexample to the Gouvea--Mazur conjecture

We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogues of Fourier-Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla's…

We give a counterexample to the most optimistic analogue (due to Kleshchev and Ram) of the James conjecture for Khovanov-Lauda-Rouquier algebras associated to simply-laced Dynkin diagrams. The first counterexample occurs in type A_5 for p =…

In their study of coarse entropy, W. Geller and M. Misiurewicz introduced the notion of coarse conjugacy: a version of conjugacy appropriate for dynamics on metric spaces observed from afar. They made two conjectures on coarse conjugacy…

We consider certain q-series depending on parameters (A,B,C), where A is a positive definite r times r matrix, B is an r-vector and C is a scalar, and ask when these q-series are modular forms. Werner Nahm conjectured a criterion for which…

In recent work of Bringmann, Guerzhoy, and the first author, p-adic modular forms were constructed from mock modular forms. This paper proves explicit congruences for these p-adic modular forms.

The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning…

In this work, we establish modular parameterizations for two general formulas for $\frac{1}{\pi}$ that subsume conjectural Ramanujan type formulas due to Z.-W. Sun, which have remained open since 2011. As an application of this, in a…

Akemann and Anderson made a conjecture about ``paving'' projections in finite dimensional matrix algebras which, if true, would settle the well-known Kadison-Singer problem. We falsify their conjecture by an explicit seqence of…

The Pappas-Rapoport coherence conjecture, proved by Zhu, states that the dimensions of spaces of sections of certain line bundles coincide. The two sides of the equality correspond to the line bundles on spherical Schubert varieties in the…

We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table of…

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in $\mathbb{P}^5$, and toric varieties of codimension two. After J.…

For a given Coleman family of modular forms, we construct a formal modeland prove the existence of a family of Galois representations associated to the Colemanfamily. As an application, we study the variations of Iwasawa $\lambda$- and…

An old conjecture of Durfee 1978 bounds the ratio of two basic invariants of complex isolated complete intersection surface singularities: the Milnor number and the singularity (or geometric) genus. We give a counterexample for the case of…

We investigate a conjecture of Robert Coleman concerning the module of circular distributions.

We give a new proof of the properness of the Coleman-Mazur eigencurve. The question of whether the eigencurve satisfies the valuative criterion for properness was first asked by Coleman and Mazur in 1998 and settled by Diao and Liu in 2016…

We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of even and odd Coleman maps for normalised new forms of arbitrary…

The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging…

Even though Zaremba's conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of…

In this paper we propose counterexamples to the Geometrization Conjecture and the Elliptization Conjecture.

The Breuil-M\'{e}zard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" in the moduli space of mod $p$ Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_q/\mathbb{Q}_q)$ that should govern congruences…