In 1987, Elkies proved the striking result that every elliptic curve over $\mathbb{Q}$ has infinitely many supersingular primes. Motivated by this theorem and its connection with the Lang-Trotter conjecture, we study the analogous problem for Frobenius fields of elliptic curves. For certain families of non-CM elliptic curves $E/\mathbb{Q}$ and imaginary quadratic fields $K$, we prove that there exist infinitely many primes $p$ for which the Frobenius field of $E$ at $p$ equals $K$. More precisely, letting $\pi_E(x,K)$ denote the number of such primes with $p\leq x$, we establish the unconditional bound $\pi_E(x, K)\gg_{E, K, \epsilon} (\log\log x)^{1-\epsilon}$ for every $\epsilon>0$. We also prove unconditional power-saving upper bounds for a restricted counting function associated with $\pi_E(x,K)$. The approach combines Deuring's theory of complex multiplication, properties of singular moduli, and arithmetic intersection theory on modular curves.
For each $r\in\{4,6\}$, we construct an explicit one-parameter family of elliptic curves over $\mathbb{Q}(i)$ containing infinitely many pairwise nonisomorphic curves genuinely defined over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $r$. We construct explicit $\mathbb{Q}(i)$-rational points to bound the...
Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the conjecture of Birch and Swinnerton-Dyer asserts that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$ as $x \to \infty$. In this paper...
Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $A,B,C \in \mathcal{O}_K \setminus\{0\}$. Denote by $S_K'$ the set of prime ideals of $\mathcal{O}_K$ dividing $2ABC$. Assuming two standard conjectures concerning the modularity of mod-$p$ Galois representations and the Eichler-Shimura correspond...
Let $y^2 = f(x,T)$ be a hyperelliptic curve of genus $g\geq 1$, defined over $\mathbb{Q}(T)$. We prove the existence of infinitely many imaginary hyperelliptic curves with a fixed genus $g$ having a certain rank for $5\leq r\leq 4g+2$, and a similar result for real hyperelliptic curves with a fixed genus $g$ having a c...
Lucas Chen, Joshua Im, Steven J. Miller et al.· 0 citations
This article studies a modular semistable elliptic curve $E$ over a totally real number field $F$ such that, upon base change to a totally imaginary quadratic extension $K$, it has analytic rank one. Assuming the Iwasawa main conjecture, along with a substantial number of assumptions, we prove a variant of the $p$-part...
Let $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner...
Antonio Lei, Luo-Chen Zhao· 0 citations
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