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, we formulate a similar conjectural asymptotic for smooth projective curves of genus at least 2, in which the contributions to the conjectured asymptotic come not only from the rank of the Jacobian but also from the Sato--Tate group of the curve. The key analytic input in formulating our conjecture is a conjecture due to Kurokawa (2012) on the convergence of Euler products of entire $L$-functions on the critical line. We also provide some numerical evidence for our conjecture in various cases.
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 correspondence over number fields, we study the asymptotic behavior of the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ over $K$. Using the modular method, we establish an asymptotic criterion in terms of the solutions of the associated $S_K'$-unit equation. As an application, we obtain asymptotic results for certain imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$. In particular, for a family of squarefree integers $d$, we determine the relevant $S_K'$-unit solutions explicitly and deduce that the generalized Fermat equation has no asymptotic solutions. Finally, we show that this family of squarefree integers has relative density $5/6$ among all squarefree positive integers.
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 certain rank for $6\leq r\leq 4g+4$. We begin by constructing such curves and prove the rank using two methods. First, we apply the generalized Nagao's conjecture, which relates the first moment and the rank of the Jacobian variety $J_\mathcal{X}(\mathbb{Q}(T))$, and that the conjecture holds for our curves, making the result unconditional. Furthermore, we explicitly construct rational points in the Mordell-Weil group and use Shioda-Tate to prove that the rank is equal to $r$.
Lucas Chen, Joshua Im, Steven J. Miller et al.· 0 citations
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.
In this second chapter of the $\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\to\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit a $3$-isogeny. Our methods are Galois-theoretic, and lead us to a thorough analysis of the Galois closure of $f:\mathbb{P}^1\to\mathbb{P}^1$. We exploit its rich geometry to prove a Selmer companionship theorem for the family $E_f$, allowing us to exhibit elements in certain Tate-Shafarevich groups which are visible in an abelian surface. We also give some dynamical and Diophantine applications of our constructions, as well as new examples of Jacobians isogenous to a power of an elliptic curve.
Let $k=\mathbb F_q$. We classify, up to two-sided $k$-M\"obius equivalence, all separable indecomposable tame exceptional maps $\mathbf P^1_k\to\mathbf P^1_k$ whose geometric Galois closure has genus one. The resulting arithmetic fixed-field forms are encoded by Frobenius-stable affine elliptic quotient data recovered from the cover; we prove a converse and an exact equivalence criterion. The same data determine branch arithmetic, geometric and arithmetic monodromy, the constant field, and behavior over every finite extension. In particular, over each $\mathbb F_{q^r}$, permutation is equivalent to exceptionality. We obtain explicit rank-two support formulas, necessary and sufficient occurrence criteria, and exact two-sided class counts in every tame signature, including the surviving cases in characteristics $2$ and $3$. In characteristic greater than $3$, tameness is automatic. A sharp $3$-adic obstruction arises only for a stronger cubic self-endomorphism realization problem, not for the fixed-field classification itself.
For $n\geq 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n)^\circ$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible N\'eron polygons. For $n \geq 10$, we give a purely algebraic and incidence-theoretic proof that, for every field $k$ with $\mathrm{char}(k)$ not dividing $n$, there is a natural bijection between $X_1(n)^\circ(k)$ and rescaling classes of $k$-realizations of $T_n$. For $k = \mathbb{C}$, this recovers a theorem of Borisov and Roulleau. We then upgrade the field-valued correspondence to an isomorphism of schemes over $\mathbb{Z}[1/n]$. The main new ingredient is a deformation-theoretic argument which allows us to verify the isomorphism on points valued in Artinian local rings. As consequences, the modular curve $X_1(n)^\circ$ acquires a natural model over $\mathbb{Z}[1/n]$ as a matroid realization space, and, for primes $p \geq 11$, the non-representability of $T_p$ over $\mathbb{Q}$ is equivalent to the prime-order case of Mazur's celebrated theorem on rational torsion points of elliptic curves. In an appendix, we explain how to upgrade the realization space of a matroid from an affine scheme over $\mathbb{Z}$ to an affine band scheme (in the sense of Baker-Jin-Lorscheid) over $\mathbb{F}_1^{\pm}$.