Related papers: Slopes of modular forms and the Ghost conjecture
Given a moduli problem with a nice coarse moduli space, which is the most intrinsic and natural divisor on this moduli space? We describe a general method of constructing effective divisors on a large class of moduli spaces using the…
After Zagier proved that the traces of singular moduli $j(z)$ are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group…
We give an explicit formula proving Xu's Conjecture on alternating double zeta values. We also discuss the limitations of Glanois' motivic Galois descent criterion in this case, as it cannot specify the depth of the descent.
Fermionic and bosonic ghost systems are defined each in terms of a single vertex algebra which admits a one-parameter family of conformal structures. The observation that these structures are related to each other provides a simple way to…
In this paper, with the help of the $S$ function and ghost symmetry for the discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the ghost flow on its eigenfunction(adjoint eigenfunction) and the spectral…
The main result of this paper is an instance of the conjecture made by Gouvea and Mazur (Math. Res. Lett., 1995) which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k…
We investigate the stability against inhomogeneous perturbations and the appearance of ghost modes in Gauss-Bonnet gravitational theories with a non-minimally coupled scalar field, which can be regarded as either the dilaton or a…
We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such…
Crystabelline representations are representations of the absolute Galois group $G_{\mathbb{Q}_p}$ over $\mathbb{Q}_p$ that become crystalline on $G_{F}$ for some abelian extension $F/\mathbb{Q}_p$. Their relation to modular forms is that…
A conjecture of Breuil, Buzzard, and Emerton says that the slopes of certain reducible $p$-adic Galois representations must be integers. In previous work we showed this conjecture for representations that lie over certain non-subtle…
Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms.…
The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from…
Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. A ghost map is a map in the stable category of finitely generated $kG$-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper…
We compute the Suslin homology of relative curves with modulus. This result may be regarded as a modulus version of the computation of motives for curves, due to Suslin and Voevodsky.
We extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We use these results to prove many new cases…
This article computes the Galois groups of congruence covers arising in the context of certain hyperbolic triangle groups. As a consequence of this computation, the genera of the respective curves are deduced.
A new Lorentz-covariant gauge is presented for SU(3). In this gauge, both the ghosts and the gauge fields in the (4, 5, 6, 7) gauge directions acquire half-integral spin. As a result, the ghosts in these directions have the correct…
We survey old and new results on the existence of moduli spaces of semistable coherent sheaves both in algebraic and in complex geometry.
We determine conditions for the invariance of the gonality under base extension, depending on the numeric invariants of the curve. More generally, we study the Galois descent of morphisms of curves to Brauer-Severi varieties, and also of…
Sequences of traces of powers of integral matrices are characterized.The result can be used can be used in character tables of e.g. the symmetric group.