Related papers: On the Gross-Stark Conjecture

For a finite abelian group $G$ and a splitting field $K$ of $G$, let $d(G, K)$ denote the largest integer $l \in \N$ for which there is a sequence $S = g_1 \cdot ... \cdot g_l$ over $G$ such that $(X^{g_1} - a_1) \cdot ... \cdot (X^{g_l} -…

For a finite group $G$, Frobenius found a formula for the values of the function $\sum_{\mathrm{Irr} G} (\dim\, \pi)^{-s}$ for even integers $s$, where $\mathrm{Irr} G$ is the set of irreducible representations of $G$. We generalize this…

A conjecture of Le says that the Deligne polytope $\Delta_d$ is generically ordinary if $p\equiv 1\ (\!\!\bmod\ D(\Delta_d))$, where $D(\Delta_d)$ is a combinatorial constant determined by $\Delta_d$. In this paper a counterexample is given…

The leading order term for the average, over quadratic discriminants satisfying the so-called Heegner condition, of the Neron-Tate height of Heegner points on a rational elliptic curve E has been determined in [12]. In addition, the second…

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem or the $q$-Dyson constant term identity. This conjecture was proved by K\'{a}rolyi, Lascoux and Warnaar…

We prove one direction of a recently posed conjecture by Gan-Gross-Prasad, which predicts the branching laws that govern restriction from p-adic $GL_n$ to $GL_{n-1}$ of irreducible smooth representations within the Arthur-type class. We…

We prove a refined version of the celebrated Lusin type theorem for gradients by Alberti, stating that any Borel vector field $f$ coincides with the gradient of a $C^1$ function $g$, outside a set $E$ of arbitrarily small Lebesgue measure.…

Let $K$ be a real quadratic field and let $p$ be a prime number which is inert in $K$. Let $K_p$ be the completion of $K$ at $p$. In a previous paper, we constructed a $p$-adic invariant $u_C\in K_p$, and we proved a $p$-adic Kronecker…

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-\alpha}$, $0 < \alpha < 1/2$. We prove that, almost surely, for every measure-preserving system…

In 2021, J.~Agler and J.~E. McCarthy proposed a two-step programme toward the celebrated Krzy\.z conjecture. The first step is to prove an entropy conjecture for polynomials whose zeros all lie on the unit circle; the second is to establish…

With respect to the analytic-algebraic dichotomy, the theory of Siegel modular forms of half-integral weight is lopsided; the analytic theory is strong whereas the algebraic lags behind. In this paper, we capitalise on this to establish the…

Let $\Gamma$ be a countable abelian group. An (abstract) $\Gamma$-system $\mathrm{X}$ - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of $\Gamma$ - is said to be a Conze-Lesigne system if…

Let S $\subseteq$ N be a numerical semigroup with multiplicity m = min(S \ {0}), conductor c = max(N \ S) + 1 and minimally generated by e elements. Let L be the set of elements of S which are smaller than c. Wilf conjectured in 1978 that…

Investigations into and around a 30-year old conjecture of Gregory Margulis and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.

There were established the exact-order estimations of the best uniform approximations by{\psi} the trigonometrical polynoms on the $C^{\psi}_{\beta,p}$ classes of $2\pi$-periodic continuous functions $f$, which are defined by the…

We introduce a strategy to tackle some known obstructions of current approaches to the Fourier uniformity conjecture. Assuming GRH, we then show the conjecture holds for intervals of length at least $(\log X)^{\psi(X)}$, with $\psi(X)…

We propose a conjecture on special values of $ L $-functions in a function field context with positive characteristic coefficients. For $ M $ a uniformizable $ t $-motive with everywhere good reduction we conjecture a relation between the…

We prove a highly uniform version of the prime number theorem for a certain class of $L$-functions. The range of $x$ depends polynomially on the analytic conductor, and the error term is expressed in terms of an optimization problem…

The ternary Goldbach conjecture states that every odd number $n\geq 7$ is the sum of three primes. The estimation of the Fourier series $\sum_{p\leq x} e(\alpha p)$ and related sums has been central to the study of the problem since Hardy…

Graham conjectured in 1971 that for any prime $p$, any subset $S\subseteq \mathbb{Z}_p\setminus \{0\}$ admits an ordering $s_1,s_2,\dots,s_{|S|}$ where all partial sums $s_1, s_1+s_2,\dots,s_1+s_2+\dots+s_{|S|}$ are distinct. We prove this…