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 ranks from below. Kai's theorem on prime values of linear patterns over number fields provides specializations with controlled local behavior, allowing us to obtain matching upper bounds via $[1+i]$-descent. The construction extends the strategy of the author's earlier rank-$2$ paper by replacing a symmetric Gaussian-prime configuration with systems of binary linear forms satisfying several complementary square identities. In the rank-$6$ case, the support vectors attached to the three constructed points and $(0,0)$ span the self-dual Reed-Muller code $\mathrm{RM}(1,3)$, which also occurs as the kernel of the quadratic-residue Laplacian governing the Selmer group.
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...
We study function fields in one variable over the field with one element $\mathbf{F}_1$. It is proved that the Galois extensions of $\mathbf{Q}$ are isomorphic to the curves over $\mathbf{F}_1$ being understood as the Deitmar schemes. Specifically, one gets explicit formulas linking the genus and the number of cusps of...
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
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...
In this paper, we give an explicit formula for the rank and generators (up to finite index) over $\mathbb{Q}(t)$ of all non-trivial elliptic curves of the form $y^2=x^3+At^6+Bt^3+C$, which is a larger class of elliptic surfaces than the one in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024), name...
Let $E$ be an elliptic curve defined over rational numbers. Further assume $E$ has good ordinary reduction at $p=2$ and $E[4]$ is reducible as a $G_\mathbb{Q}$-representation. In this paper, we offer sufficient and necessary computational criteria for the algebraic Iwasawa $\mu$-invariant over the cyclotomic $\mathbb{Z...
Zi-Chao Lin, Mu-Lun Yin· 0 citations
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