Related papers: Explicit upper bound for the rank of J_0(q)

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup of $G$. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of $G$ in terms of the size of the abelianization of $P$. To do…

For a prime $\mathfrak{p} \subseteq \mathbb{F}_{q}[T]$ and a positive integer $r$, we consider the generalised Jacobian $J_{0}(\mathfrak{n})_{\mathbf{m}}$ of the Drinfeld modular curve $X_{0}(\mathfrak{n})$ of level…

We give an algorithm to explicitly determine all elements of the $q$-torsion (for $q$ an odd prime) of the Brauer group of an elliptic curve over any base field of characteristic different from $q$, containing a primitive $q$-th root of…

In this paper, we obtain some estimations of the saddle order which is the sole topological invariant of the non-integrable resonant saddles of planar polynomial vector fields of arbitrary degree $n$. Firstly, we prove that, for any given…

In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We show that the analytic rank of $E$ over the cyclotomic extension $\mathbb{Q}(e^{2\pi i/q})$ is bounded above by $q^{45/52+\varepsilon}$, as $q\to \infty$ through the primes. This…

We use methods for computing Picard numbers of reductions of K3 surfaces in order to study the decomposability of Jacobians over number fields and the variance of Mordell-Weil ranks of families of Jacobians over different ground fields. For…

By applying the Euler--Rayleigh methods to a specific representation of the Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, we derive upper and lower bound for…

We study the problem of $d$-gonality of the modular curve $X_0(N)$. As a result, we can give an upperbound of the level $N$ by means of $d$. This generalizes Ogg's result on hyperelliptic modular curves ($d = 2$). As a corollary of this…

In this note, we obtain an exact formula for the $q$-adic valuation of $\sigma_k(n)$ where $q$ is an odd prime, allowing us to derive an explicit upper bound which is asymptotically better than the previous bound obtained by Zhao when $n$…

We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $\mathbb{F}_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$. When $q$ is a…

Let $M_{q}(k)$ be the maximum length of MDS codes with parameters $q,k$. In this paper, the properties of $M_{q}(k)$ are studied, and some new upper bounds of $M_{q}(k)$ are obtained. Especially we obtain that $M_{q}(q-1)\leq…

We develop refined methods to effectively bound the dimension of Bloch-Kato Selmer groups associated to the higher Chow group $\mathrm{CH}^2(J,1)$, where $J$ is the Jacobian of a hyperelliptic curve $X$. This extends the recent work of…

We carry out a complete birational classification of the degree g universal Jacobian P_g over the moduli space of curves, highlighting the transition cases g=10, 11. The universal Jacobian is unirational when g<10, has Kodaira dimension…

Let $\chi$ be a Dirichlet character modulo a prime~$p$. We give explicit upper bounds on $q_1<q_2<\dots<q_n$, the $n$ smallest prime nonresidues of $\chi$. More precisely, given $n_0$ and $p_0$ there exists an absolute constant…

We prove that there exist infinitely many elliptic curves over \Q with given modular invariant, and rank >=2. Furthermore, there exist infinitely many elliptic curves over $\Q$ with invariant equal at 0 (resp. 1728) and rank >=6 (resp.…

We continue the study of the Drinfeld double of the Jordan plane, denoted by $\mathcal D$ and introduced in arXiv:2002.02514. The simple finite-dimensional modules were computed in arXiv:2108.13849; it turns out that they factorize through…

We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related…

In this paper the family of elliptic curves over \Q given by the equation E_{p}: Y^2=(X-p)^3+X^3+(X+p)^3 where p is a prime number, is studied. It is shown that the maximal rank of the elliptic curves is at most 3 and some conditions under…

We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average…