Let $E$ be an elliptic curve over $\mathbb{Q}_p$. We study the field $\mathbb{Q}_p(E[p])$ generated by the $p$-torsion points of $E$. When $E$ has good reduction, we determine not only $\mathbb{Q}_p(E[p])$ but also the field $\mathbb{Q}_p(P)$ for every point $P\in E[p]$. This classification yields criteria for the existence of $p$-torsion over unramified and Lubin--Tate extensions of $\mathbb{Q}_p$. As a global application, let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be an odd prime of good reduction. Define $f_p(E)$ to be the multiplicative order of $a_p(E)$ modulo $p$ in the ordinary case, and set $f_p(E)=p^2-1$ in the supersingular case. We prove that $E(K)[p]=0$ for every number field $K/\mathbb{Q}$ with $[K:\mathbb{Q}]<f_p(E)$. When $E$ has bad reduction, we describe the ramified part of $\mathbb{Q}_p(E[p])/\mathbb{Q}_p$. For every reduction type, we determine the maximal upper ramification break of $\mathbb{Q}_p(E[p])/\mathbb{Q}_p$.
Let $p \geq 5$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E/K$ be an elliptic curve with good reduction. Let $F_n$ denote the $n$-division polynomial of $E$. Silverman proved that if the reduction is ordinary, then for every $P \in E(K) \setminus \hat{E}(K)$ and a suitable power $q$ of $p$, t...
For a smooth projective curve $C/\mathbb{Q}$ of genus $\geq 2$ and $L/\mathbb{Q}$ an extension, we write $C(L)_{\text{new}}=\{P\in C(L):\mathbb{Q}(P)=L\}$. Recent work of Khawaja and Siksek conjectures that this set is empty for $100\%$ of degree $n$ number fields $L$, when ordered by absolute discriminant. Moreover, t...
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 $K$ be an imaginary quadratic number field, let $p$ be an odd prime, and let $G=G_{\varnothing}(K)(p)$ be the Galois group of the maximal everywhere unramified pro-$p$ extension of $K$. To each mod-$p$ character $x$ of $G$ we associate a linear map $D_x$ from $\mathrm{Cl}(K)[p]$ to $\mathrm{Cl}(K)/p$; a formula of...
We develop Hachimori's study on the unramified Iwasawa modules using extensions with restricted $p$-ramification. Let $K$ be an algebraic number field, and $p$ a prime number which splits into two distinct primes $\mathfrak{p}$, $\mathfrak{p}'$ in $K$. Assume that $K$ and $p$ satisfies several (somewhat strict) conditi...
Let $E_{A,B}\colon y^2 +Axy + By = x^3$ be an elliptic curve over $\mathbb{Q}$. The $3$-torsion point $(0,0)$ induces a $3$-isogeny $\phi\colon E_{A,B}\to E_{A,B}'$. Assuming $E_{A,B}$ has good reduction at $3$, we construct an explicit $(m+t)\times m$ matrix $M_{A,B}'$ over $\mathbb{F}_3$, whose kernel encodes the dua...
Ariel Weiss, Dong-Chen Zou· 0 citations
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