Related papers: Sur le rang de J_0(q)

Let $\mathcal{X}$ be a Riemann surface of genus $g>0$ defined over a number field $K$ which is a degree $d$-covering of $\mathbb{P}^1_K$. In this paper we show the existence of infinitely many linearly disjoint degree $d$-extensions $L/K$…

Fix an elliptic curve $E/\Q$, and assume the generalized Riemann hypothesis for the $L$-function $ L(E_D, s) $ for every quadratic twist $E_D$ of $E$ by $D\in\Z$. We combine Weil's explicit formula with techniques of Heath-Brown to derive…

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set…

Let $E_{/\mathbb{Q}}$ be an elliptic curve with rank $E(\mathbb{Q})=0$. Fix an odd prime $p$, a positive integer $n$ and a finite abelian extension $K/\mathbb{Q}$ with rank $E(K) = 0$. In this paper, we show that there exist infinitely many…

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…

Using Laurent decomposition and Mergelyan's theorem combined with Baire's category theorem, we prove generic nowhere differentiability on the distinguished boundary of functions of analytic type on products of planar domains bounded by…

We prove a structural upper bound on the $p$-adic valuation of denominators of rationals belonging to a missing-digit set $K_{m,D}$, generalizing a key step in recent work of Lin, Wu, and Yang [arXiv:2603.24614] on reciprocals of…

A central problem in Diophantine geometry is to uniformly bound the number of $K$-rational points on a smooth curve $X/K$ in terms of $K$ and its genus $g$. A recent paper by Stoll proved uniform bounds for the number of $K$-rational points…

For the Hamming graph $H(n,q)$, where a $q$ is a constant prime power and $n$ grows, we construct perfect colorings without non-essential arguments such that $n$ depends exponentially on the off-diagonal part of the quotient matrix. In…

Building on work in \cite{AB24} on the Riemann zeta function at height $T$ off the critical line, we prove an unconditional lower bound on the critical line for real large deviations of the order $V\sim\alpha\log\log T$ for any $\alpha>0.$…

Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of $L$-functions lie on the critical line with the real part $1/2$. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level…

We show, for any $q\ge 3$ and distinct reduced residues $a,b \pmod q$, the existence of certain hypothetical sets of zeros of Dirichlet $L$-functions lying off the critical line implies that $\pi(x;q,a)<\pi(x;q,b)$ for a set of real $x$ of…

We generalize a result by Alberti, showing that, if a first-order linear differential operator $\mathcal{A}$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure $\mu$ satisfying…

The Riemann Zeta-Function is the most studied L-function; it's zeroes give information about the prime numbers. We can associate L-functions to a wide array of objects, and in general, the zeroes of these L-functions give information about…

Let $E/k$ be a non-isotrivial elliptic curve over a global function field $k$ of characteristic $p>3$, and $G\subset \mathrm{Gal}(k^{\mathrm{sep}}/k)$ be a topologically finitely generated subgroup. We prove that if $E/k$ has analytic rank…

Let $C$ be a smooth projective irreducible curve defined over a finite field $\mathbb{F}_q$ and $K=\mathbb{F}_q(C)$. Let $A\subset K$ be the ring of functions regular outside a fixed place $\infty$ of $K$. Let…

Given an $L$-function, one of the most important questions concerns its vanishing at the central point; for example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing there of an elliptic curve $L$-function equals…

We prove the following statement about any Siegel modular form $F$ of degree $n$ and arbitrary odd level $N$ on the group $\Gamma_{0}^{(n)}(N)$. Let $A(F,T)$ denote the Fourier coefficients of $F$ and write $T=(T(i,j))$. Suppose that $F$…

We show that the set of Julia limiting directions of a transcendental-type $K$-quasiregular mapping $f:\mathbb{R}^n\to \mathbb{R}^n$ must contain a component of a certain size, depending on the dimension $n$, the maximal dilatation $K$, and…

We prove logarithmic Sobolev inequality for measures $$ q^n(x^n)=\text{dist}(X^n)=\exp\bigl(-V(x^n)\bigr), \quad x^n\in \Bbb R^n, $$ under the assumptions that: (i) the conditional distributions $$ Q_i(\cdot| x_j, j\neq i)=\text{dist}(X_i|…