Let $F$ be a totally real Galois number field of degree $g\leq5$ and let $\ell$ be an odd prime. Let $E/F$ be an elliptic curve with an $F$-rational point of order $\ell$. Under explicit arithmetic and local hypotheses, together with an explicit non-vanishing condition modulo $\ell$, we prove that there are $\gg_{F,E,\ell}\frac{X^{1/(2g)}}{\log X}$ totally negative square classes $d\in F^\times/(F^\times)^2$ with $\left|\mathrm{N}_{F/\mathbb{Q}}\bigl(D(F(\sqrt{d})/F)\bigr)\right|<X $ for which $\operatorname{Sel}_\ell(E^d,F)$ is trivial. Such twists have rank zero and trivial $\ell$-part of the Shafarevich--Tate group. The condition is verifiable by a finite computation, which we carry out in an example. We lift the method of James and Ono from $\mathbb{Q}$ to the totally real setting, combining a theorem of Morrow, which relates the Selmer group of such a twist to the class group of the CM extension $F(\sqrt{d})$, with an indivisibility theorem of Takai for relative class numbers. Takai twists only by quadratic Hecke characters, whereas Morrow's conditions are local; we extend the twisting argument to primitive quadratic residue-class characters to make the two results compatible.
Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field and let $\ell$ be a prime invertible in the field. For every constructible $\mathbb{F}_{\ell}$-sheaf $F$ on $A$ we prove that there is an integer $e$, depending on $F$, such that the pullback map $[\ell^{e}]^{*}\colon\mathrm{H}^{i}(A,[\el...
Ashutosh Roy Choudhury, K. V. Shuddhodan· 0 citations
Let $n\geq 1$ and let $\ell$ be an odd prime. Let $G=\mathrm{PGSp}_{2n}(\mathbb{F}_\ell)$ or $\mathrm{GSp}_{2n}(\mathbb{F}_\ell)$, and fix a faithful transitive permutation representation $\pi:G\longrightarrow S_d$. We study degree-$d$ number fields whose Galois closures have Galois group $G$ and whose associated permu...
Let $\ell$ be an odd prime number. Let $E$ be a non-isotrivial elliptic curve defined over the global function field $K = \mathbb{F}_q(t)$ such that $(q, 6) = 1$ and $q \equiv 1 \mod \ell$. We show that as $q$ grows arbitrarily large, both analytic and algebraic ranks of $E$ do not grow upon base change to $100\%$ of $...
For every even Galois representation in a family of prime conductor $\ell$, we prove that if $k \in \mathbb{Z}_{\geq 1}$ is maximal such that \[\ell \equiv 1 \bmod 3^k\] then the even mod $3$ deformation ring $R_S \otimes_{\mathbb{Z}_3} \mathbb{F}_3$ of level $S=\{\ell,3\}$ does not have a quotient isomorphic to $\math...
Fix a prime $\ell$. Let $K = \mathbb{F}_q(t)$ be a global function field such that $\gcd(q,6) = 1$ and $q \equiv 1 \pmod \ell$. Let $E$ be a non-isotrivial elliptic curve over $K$. Given a fixed monic polynomial $Q$ over $\mathbb{F}_q$, and assuming some mild conditions on $E$, we show that the rank of $E$ does not cha...
Let $C/\mathbb{Q}$ be a Picard curve and let $\ell\equiv1\pmod3$ be a prime such that the $\lambda$-torsion representation of the Jacobian of $C$ is reducible for a prime $\lambda$ of $\mathbb{Q}(\zeta_3)$ above $\ell$. We describe the constraints imposed by complex conjugation on the $2$-dimensional subquotient $\rho$...
Xi-Yao Chen, S. Chidambaram, Andrew Coursin et al.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
