This paper presents various transcendence results in the ring of integers modulo infinitely large primes $\mathcal {A}$. In the ring $\mathcal {A}$, one can consider two notions of transcendence. One is based on the notion of finite algebraic numbers introduced by Rosen, while the other is transcendence in the naive sense. It is known that transcendence in the latter sense automatically implies transcendence in the former sense. In this paper, we strengthen results due to Anzawa and Funakura and to Luca and Zudilin by removing some of their assumptions and, in some cases, upgrading them to statements of naive transcendence. We also present several examples of naive transcendental numbers that do not seem to have appeared previously in the literature. Although we are not able to establish naive transcendence for certain numbers, we prove the irrationality of numbers such as $\log _{\mathcal {A}}(2)$ under the ABC conjecture.

Division polynomials associated with an elliptic curve $E/K$ are polynomials $\phi _n, \psi _n^2$ that arise from the sequence of points $\{nP\}_{n \in \mathbb {N}}$ on this curve. If one wishes to study $\mathbb {Z}$-linear combination of points on $E(K)$, we can use net polynomials $\Phi _{\boldsymbol {v}}, \Psi _{\boldsymbol {v}}^2$ which are higher-dimensional analogs of division polynomials. It turns out that they are also elliptic nets, an $n$-dimensional array with values in $K$ satisfying the same nonlinear recurrence relation that division polynomials do as well. Now assume further that the elliptic curve $E/K$ has complex multiplication by an order of a quadratic imaginary field $F \subseteq K$, we will prove a formula for the common valuation of $\Phi _{\boldsymbol {v}}$ and $\Psi _{\boldsymbol {v}}^2$ associated with multiples of points by elements of an order in $F$. As an application, we will use the formula to show that elliptic divisibility sequences associated with multiples of points indexed by elements of an order also satisfy a recurrence relation when indexed by elements of an order, subject to certain conditions on the indices.

In this article, we study Iwasawa theory for the Selmer group of an elliptic curve over a certain non-commutative p-adic Lie extension whose Galois group is isomorphic to an open subgroup of $\mathrm {PGL}_2(\mathbb {Z}_p)$. Unlike the classes of extension usually considered in non-commutative Iwasawa theory, this extension does not contain the cyclotomic $\mathbb {Z}_p$-extension. We give a formula for its Euler characteristic in this case. We also show that the dual Selmer group contains no nonzero pseudonull submodules, a result analogous to a theorem of Ochi and Venjakob for $\mathrm {GL}_2$-extensions. Finally, we provide an explicit formula computing the change in Iwasawa $\mu $-invariants of dual Selmer groups when two elliptic curves are isogenous; this in the same spirit as that of Howson’s result for the $\mathrm {GL}_2$-extension which extends results in the commutative case by Schneider and Perrin-Riou.

In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{{\mathbb{Q}}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in {\mathbb{Z}}^+$, we wish to determine the least degree of a point on the modular curve $X_1(N)$ associated to any $E \in \mathcal{E}$. We consider the cases where $\mathcal{E}$ is rational, i.e., contains an elliptic curve with rational j-invariant, or where $\mathcal{E}$ consists of elliptic curves with complex multiplication (CM). If $N=\ell^k$ is a power of a single prime, we give a complete characterisation upon restricting to points of odd degree, and also in the case where $\mathcal{E}$ is CM. We include various partial results in the more general setting.

We study the boundedness of the Mordell–Weil rank and the growth of the v-primary part of the Tate–Shafarevich group of p-supersingular abelian varieties of $\mathrm {GL}_2$-type with real multiplication over $\mathbb Z_p$-extensions of number fields, where v is a prime lying above p. Building on the work of Iovita and Pollack in the case of elliptic curves, under precise ramification and splitting conditions on p, we construct explicit systems of local points using the theory of Lubin–Tate formal groups. We then define signed Coleman maps, which in turn allow us to formulate and analyse signed Selmer groups. Assuming these Selmer groups are cotorsion, we prove that the Mordell–Weil groups are bounded over any subextensions of the ${\mathbb Z}_p$-extension and provide an asymptotic formula for the growth of the v-primary part of the Tate–Shafarevich groups. Our results extend those of Kobayashi, Pollack, and Sprung on p-supersingular elliptic curves.

This paper addresses the prediction of positive rank for elliptic curves without the need to find a point of infinite order or compute L-functions. While the most common method relies on parity conjectures, a recent technique introduced by Dokchitser, Wiersema and Evans predicts positive rank based on the value of a certain product of Tamagawa numbers, raising questions about its relationship to parity. We show that their method is a subset of the parity conjectures approach: whenever their method predicts positive rank, so does the use of parity conjectures. To establish this, we extend previous work on Brauer relations and regulator constants to a broader setting involving combinations of permutation modules known as K-relations. A central ingredient in our argument is demonstrating a compatibility between Tamagawa numbers and local root numbers.

In the 17th century, a letter from Fermat to Mersenne considered the problem of finding the Pythagorean triples in which the hypotenuse and the sum of the other two sides are squares. We call this question the Fermat–Mersenne problem. Number-theoretic solutions of the problem are known, but they are complicated. We solve the Fermat–Mersenne problem by using a geometric approach that provides new insight. The argument involves stereographically projecting points onto an elliptic curve and applying complex multiplication on that curve.

Let p and q be two primes with $(p,q)\equiv (1,5)$ or $(7,3) \pmod 8$. Lagrange [‘Nombres congruents et courbes elliptiques’, Séminaire Delange-Pisot-Poitou. Théorie des Nombres 16(1) (1974–1975), Article no. 16] and Qin [‘Congruent numbers, quadratic forms and $K_2$’, Math. Ann. 383(3–4) (2022), 1647–1686] showed that if $(\frac {q}{p})=-1$, then $2pq$ is not a congruent number. By using Qin’s method, we prove that if $(p,q)\equiv (1,5) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-pq)\not \equiv p-1 \pmod {16}$, then $2pq$ is not a congruent number; if $(p,q)\equiv (7,3) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-2pq)\not \equiv p+1 \pmod {16}$, then $2pq$ is not a congruent number. Here, $h(-d)$ denotes the class number of the imaginary quadratic field $\mathbb {Q}(\sqrt {-d})$.

The notion of $\theta $-congruent numbers generalises the classical congruent number problem. A positive integer n is $\theta $-congruent if it is the area of a rational triangle with an angle $\theta $ whose cosine is rational. Das and Saikia [‘On $\theta $-congruent numbers over real number fields’, Bull. Aust. Math. Soc. 103(2) (2021), 218–229] established criteria for numbers to be $\theta $-congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $\theta $-congruent and properly $\theta $-congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to six, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree six and examine the exceptional cases $n=1, 2, 3$ and $6$.

Under the Generalised Riemann Hypothesis (GRH), any element in the multiplicative group of a number field K that is globally primitive (i.e., not a perfect power in $K^*$) is a primitive root modulo a set of primes of K of positive density.

For elliptic curves $E/K$ that are known to have infinitely many primes ${\mathfrak{p}}$ of cyclic reduction, possibly under GRH, a globally primitive point $P\in E(K)$ may fail to generate any of the point groups $E(k_{\mathfrak{p}})$. We describe this phenomenon in terms of an associated Galois representation $\rho_{E/K, P}\,:\,G_K\to\mathrm{GL}_3({\widehat {{\mathbf{Z}}}})$, and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive.

Fix a prime number p. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2 and 3, and containing the primitive pth root of unity $\mu_p$. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Poonen and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erdös–Kac theorem, and the geometric ergodicity of Markov chains.

Let F be a totally real field. Let $\mathsf {A}$ be a simple modular self-dual abelian variety defined over F. We study the growth of the corank of Selmer groups of $\mathsf {A}$ over $\mathbb {Z}_p$-extensions of a complex multiplication (CM) extension of F. We propose an extension of Mazur’s growth number conjecture for elliptic curves to this new setting. We provide evidence supporting an affirmative answer by studying special cases of this problem, generalising previous results on elliptic curves and imaginary quadratic fields.

Let K be an imaginary quadratic field, and let $E/\mathbb {Q}$ be an elliptic curve with complex multiplication by $\mathcal {O}_K$. Let $K_\infty /K$ be the anticyclotomic $\mathbb {Z}_p$-extension of K and $K_n$ be the intermediate layers. Under additional assumptions on Kobayashi’s signed Selmer groups, we prove an asymptotic formula for .

Let E be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\mathbb {Q}$ has open image, and in particular, there is a minimal natural number $C_E$ such that the mod $\ell $ representation ${\bar {\rho }}_{E,\ell }$ is surjective for any prime $\ell> C_E$. Assuming the Generalized Riemann Hypothesis, Mayle–Wang gave explicit bounds for $C_E$ which are logarithmic in the conductor of E and have explicit constants. The method is based on using effective forms of the Chebotarev Density Theorem together with the Faltings–Serre method, in particular, using the “deviation group” of the $2$-adic representations attached to two elliptic curves.

By considering quotients of the deviation group and a characterization of the images of the $2$-adic representation $\rho _{E,2}$ by Rouse and Zureick–Brown, we show in this article how to further reduce the constants in Mayle–Wang’s results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.

In the early 2000s, Ramakrishna asked the question: for the elliptic curve

\[E\;:\; y^2 = x^3 - x,\]

$a_p(E)$

$E/\mathbb{Q}$

The cyclicity and Koblitz conjectures ask about the distribution of primes of cyclic and prime-order reduction, respectively, for elliptic curves over $\mathbb {Q}$. In 1976, Serre gave a conditional proof of the cyclicity conjecture, but the Koblitz conjecture (refined by Zywina in 2011) remains open. The conjectures are now known unconditionally “on average” due to work of Banks–Shparlinski and Balog–Cojocaru–David. Recently, there has been a growing interest in the cyclicity conjecture for primes in arithmetic progressions (AP), with relevant work by Akbal–Güloğlu and Wong. In this article, we adapt Zywina’s method to formulate the Koblitz conjecture for AP and refine a theorem of Jones to establish results on the moments of the constants in both the cyclicity and Koblitz conjectures for AP. In doing so, we uncover a somewhat counterintuitive phenomenon: On average, these two constants are oppositely biased over congruence classes. Finally, in an accompanying repository, we give Magma code for computing the constants discussed in this article.

Given an elliptic curve $ E $ over $ \mathbb {Q} $ of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character $ \chi $ of order $ q $, and in many cases its associated special L-value $ \mathscr {L}(E, \chi ) $ is known to be integral after normalizing by certain periods. This article determines the precise value of $ \mathscr {L}(E, \chi ) $ in terms of Birch–Swinnerton-Dyer invariants when $ q = 3 $, classifies their asymptotic densities modulo $ 3 $ by fixing $ E $ and varying $ \chi $, and presents a lower bound on the $ 3 $-adic valuation of $ \mathscr {L}(E, 1) $, all of which arise from a congruence of modular symbols. These results also explain some phenomena observed by Dokchitser–Evans–Wiersema and by Kisilevsky–Nam.

We explore the relationship between (3-isogeny induced) Selmer group of an elliptic curve and the (3 part of) the ideal class group, over certain non-abelian number fields.

Let $e$ and $q$ be fixed co-prime integers satisfying $1\lt e\lt q$. Let $\mathscr {C}$ be a certain family of deformations of the curve $y^e=x^q$. That family is called the $(e,q)$-curve and is one of the types of curves called plane telescopic curves. Let $\varDelta$ be the discriminant of $\mathscr {C}$. Following pioneering work by Buchstaber and Leykin (BL), we determine the canonical basis $\{ L_j \}$ of the space of derivations tangent to the variety $\varDelta =0$ and describe their specific properties. Such a set $\{ L_j \}$ gives rise to a system of linear partial differential equations (heat equations) satisfied by the function $\sigma (u)$ associated with $\mathscr {C}$, and eventually gives its explicit power series expansion. This is a natural generalisation of Weierstrass’ result on his sigma function. We attempt to give an accessible description of various aspects of the BL theory. Especially, the text contains detailed proofs for several useful formulae and known facts since we know of no works which include their proofs.

We show that for $5/6$-th of all primes p, Hilbert’s 10th problem is unsolvable for the ring of integers of $\mathbb {Q}(\zeta _3, \sqrt [3]{p})$. We also show that there is an infinite set S of square-free integers such that Hilbert’s 10th problem is unsolvable over the ring of integers of $\mathbb {Q}(\zeta _3, \sqrt {D}, \sqrt [3]{p})$ for every $D \in S$ and for every prime $p \equiv 2, 5\ \pmod 9$. We use the CM elliptic curves $y^2=x^3-432 D^2$ associated with the cube-sum problem, with D varying in suitable congruence class, in our proof.