For an elliptic curve E defined over a number field K and $L/K$ a Galois extension, we study the possibilities of the Galois group Gal$(L/K)$, when the Mordell–Weil rank of $E(L)$ increases from that of $E(K)$ by a small amount (namely 1, 2, and 3). In relation with the vanishing of corresponding L-functions at $s=1$, we prove several elliptic analogues of classical theorems related to Artin’s holomorphy conjecture. We then apply these to study the analytic minimal subfield, first introduced by Akbary and Murty, for the case when order of vanishing is 2. We also investigate how the order of vanishing changes as rank increases by 1 and vice versa, generalizing a theorem of Kolyvagin.

In this note, we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials $\psi _n$ and $\phi _n$ associated with a sequence $\{nP\}_{n\in \mathbb {N}}$ of points on an elliptic curve $E$ defined over a discrete valuation field $K$. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.

We study the discriminants of the minimal polynomials $\mathcal {P}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive $n\equiv 11\pmod {24}$. We show that $\Delta (\mathcal {P}_n)$ divides $\Delta (H_n)$, where $H_n$ is the ring class polynomial, with quotient a perfect square and determine the sign of $\Delta (\mathcal {P}_n)$ based on the ideal class group structure of the order of discriminant $-n$. We also show that the discriminant of the number field generated by $j({(-1+\sqrt {-n})}/{2})$, where j is the j-invariant, divides $\Delta (\mathcal {P}_n)$. Moreover, using Ye’s computation of $\log|\Delta(H_n)|$ [‘Revisiting the Gross–Zagier discriminant formula’, Math. Nachr. 293 (2020), 1801–1826], we show that 3 never divides $\Delta(H_n)$, and thus $\Delta(\mathcal{P}_n)$, for all squarefree $n\equiv11\pmod{24}$.

Let E be an elliptic curve with positive rank over a number field K and let p be an odd prime number. Let $K_{\operatorname {cyc}}$ be the cyclotomic $\mathbb {Z}_p$ -extension of K and $K_n$ its nth layer. The Mordell–Weil rank of E is said to be constant in the cyclotomic tower of K if for all n, the rank of $E(K_n)$ is equal to the rank of $E(K)$ . We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in this sense. We then indicate the potential applications to Hilbert’s tenth problem for number rings.

Let $A$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. We prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves.

Let $g \geq 1$ be an integer and let $A/\mathbb Q$ be an abelian variety that is isogenous over $\mathbb Q$ to a product of g elliptic curves defined over $\mathbb Q$ , pairwise non-isogenous over $\overline {\mathbb Q}$ and each without complex multiplication. For an integer t and a positive real number x, denote by $\pi _A(x, t)$ the number of primes $p \leq x$ , of good reduction for A, for which the Frobenius trace $a_{1, p}(A)$ associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that $\pi _A(x, 0) \ll _A x^{1 - \frac {1}{3 g+1 }}/(\operatorname {log} x)^{1 - \frac {2}{3 g+1}}$ and $\pi _A(x, t) \ll _A x^{1 - \frac {1}{3 g + 2}}/(\operatorname {log} x)^{1 - \frac {2}{3 g + 2}}$ if $t \neq 0$ . These bounds largely improve upon recent ones obtained for $g = 2$ by Chen, Jones, and Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for $g=1$ by Murty, Murty, and Saradha, combined with a refinement in the power of $\operatorname {log} x$ by Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying $|a_{1, p}(A)|>p^{\frac {1}{3 g + 1} - \varepsilon }$ for any fixed $\varepsilon>0$ .

We show that the conjecture of [27] for the special value at $s=1$ of the zeta function of an arithmetic surface is equivalent to the Birch–Swinnerton–Dyer conjecture for the Jacobian of the generic fibre.

Let E be an elliptic curve defined over a number field F with good ordinary reduction at all primes above p, and let $F_\infty $ be a finitely ramified uniform pro-p extension of F containing the cyclotomic $\mathbb {Z}_p$ -extension $F_{\operatorname {cyc}}$ . Set $F^{(n)}$ be the nth layer of the tower, and $F^{(n)}_{\operatorname {cyc}}$ the cyclotomic $\mathbb {Z}_p$ -extension of $F^{(n)}$ . We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $\lambda $ -invariant of the Selmer group over $F^{(n)}_{ \operatorname {cyc}}$ as $n\rightarrow \infty $ . This method has its origins in work of A. Cuoco, who studied $\mathbb {Z}_p^2$ -extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases.

Let $E/\mathbb {Q}$ be an elliptic curve. For a prime p of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the x-coordinate of a point of maximal order in the group $E(\mathbb {F}_p)$ . We prove unconditionally that $r(E,p)> 0.72\log \log p$ for infinitely many p, and $r(E,p)> 0.36 \log p$ under the assumption of the Generalized Riemann Hypothesis. These can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime.

In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.

In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient$\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$. Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$-Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$. The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

We compute the $g=1$, $n=1$ B-model Gromov–Witten invariant of an elliptic curve $E$ directly from the derived category $\mathsf{D}_{\mathsf{coh}}^{b}(E)$. More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic $\mathscr{A}_{\infty }$ model of $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ described by Polishchuk. This is the first non-trivial computation of a positive-genus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov–Witten invariants of a symplectic 2-torus computed by Dijkgraaf.

Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$-dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb {P}^{2}$. In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$-dimensional quadratic AS-regular algebras that cannot be written as a twist of a Calabi–Yau AS-regular algebra by a graded algebra automorphism.

Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in terms of $g$ and the size of $S$. In the second part we assume that $L$ is the function field of a hyperelliptic curve $C_{A}:s^{2}=A(t)$, where $A(t)$ is a square-free $k$-polynomial of odd degree. If $\infty$ is the place of $L$ associated to the point at infinity of $C_{A}$, then we prove that the set of separable $\{\infty \}$-points can be bounded solely in terms of $g$ and does not depend on the Mordell–Weil group $E(L)$. This is done by bounding the number of separable integral points over $k(t)$ on elliptic curves of the form $E_{A}:A(t)y^{2}=f(x)$, where $f(x)$ is a polynomial over $k$. Additionally, we show that, under an extra condition on $A(t)$, the existence of a separable integral point of ‘small’ height on the elliptic curve $E_{A}/k(t)$ determines the isomorphism class of the elliptic curve $y^{2}=f(x)$.

We establish an error term in the Sato–Tate theorem of Birch. That is, for $p$ prime, $q=p^{r}$ and an elliptic curve $E:y^{2}=x^{3}+ax+b$, we show that

$$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$

$I\subseteq [0,\unicode[STIX]{x1D70B}]$

$\unicode[STIX]{x1D703}_{a,b}$

$2\sqrt{q}\cos \unicode[STIX]{x1D703}_{a,b}=q+1-E(\mathbb{F}_{q})$

$\unicode[STIX]{x1D707}_{ST}(I)$

$I$

In this paper, we prove some conjectures of K. Stolarsky concerning the first and third moments of the Beatty sequences with the golden section and its square.