Related papers: Ribet's conjecture for Eisenstein maximal ideals

In the monograph arXiv:2108.03453, we define the notion of a unipotent representation of a complex reductive group. The representations we define include, as a proper subset, all special unipotent representations in the sense of…

Let N be a square-free positive integer and let f be a newform of weight 2 on \Gamma_0(N). Let A denote the abelian subvariety of J_0(N) associated to f and let m be a maximal ideal of the Hecke algebra T that contains Ann_T(f) and has…

Let $N$ be a non-squarefree integer such that the quotient $X_0(N)^*$ of the modular curve $X_0(N)$ by the full group of Atkin-Lehner involutions has positive genus. Elkies conjectures that the rational points on $X_0(N)^*$ are only cusps…

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 =…

An ideal $I$ of a commutative ring $R$ is said to be of linear type when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if $G$…

Zassenhaus Conjecture for torsion units states that every augmentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of rational group algebra QG. This conjecture has been…

A shape optimization program is developed for the ratio of Riesz capacities $\text{Cap}_q(K)/\text{Cap}_p(K)$, where $K$ ranges over compact sets in $\mathbb{R}^n$. In different regions of the $pq$-parameter plane, maximality is conjectured…

For a positive integer $N$, let $J_0(N)$ be the Jacobian of the modular curve $X_0(N)$. In this paper we completely determine the structure of the rational cuspidal subgroup of $J_0(N)$ when the largest perfect square dividing $N$ is either…

Let $\frak{p}$ and $\frak{q}$ be two distinct prime ideals of $\mathbb{F}_q[T]$. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve $X_0(\frak{p}\frak{q})$ to compare the rational torsion subgroup of the Jacobian…

A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successfully applied to several hard special cases…

A well known conjecture asserts that a cubic fourfold $X$ whose transcendental cohomology $T_X$ can not be realized as the transcendental cohomology of a $K3$ surface is irrational. Since the geometry of cubic fourfolds is intricately…

In the early 1970s, Andrew Ogg made several conjectures about the rational torsion points of elliptic curves over $\mathbb{Q}$ and the Jacobians of modular curves. These conjectures were proved shortly after by Barry Mazur as a consequence…

The Serre conjecture II predicts that every torsor under a semisimple, simply connected, algebraic group over a field of cohomological dimension at most 2 and of degree of imperfection at most 1 has a rational point. We generalize this…

Consider a smooth, geometrically irreducible, projective curve of genus $g \ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that…

We consider the generalized Jacobian $\widetilde{J}_0(N)$ of a modular curve $X_0(N)$ with respect to a reduced divisor given by the sum of all cusps on it. When $N$ is a power of a prime $\geq 5$, we exhibit that the group of rational…

L\'evai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all…

Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an…

Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over $\mathbb{Q}$ and its Jacobian variety $J_0(3p)$ over $\mathbb{Q}$. Let $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ be the group of rational torsion points on $J_0(3p)$…

Let $k$ be a number field, $f(x)\in k[x]$ a polynomial over $k$ with $f(0)\neq 0$, and $\O_{k,S}^*$ the group of $S$-units of $k$, where $S$ is an appropriate finite set of places of $k$. In this note, we prove that outside of some natural…

The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn's lemma, and thus pose a challenge from a computational point of view. Giving a…