We show that if F is $\mathbb{Q}$ or a multiquadratic number field, $p\in\left\{{2,3,5}\right\}$, and $K/F$ is a Galois extension of degree a power of p, then for elliptic curves $E/\mathbb{Q}$ ordered by height, the average dimension of the p-Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K-rank of elliptic curves $E/\mathbb{Q}$ for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F.

The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p, as $E/\mathbb{Q}$ varies, the difference in dimension between the Galois fixed space in the p-Selmer group of $E/K$ and the p-Selmer group of $E/F$ has bounded average.

We discuss the p-adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the p-adic domain. These results are extensions of the p-adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3] and also generalise a result of Bertrand to higher dimensions [‘Sous-groupes à un paramètre p-adique de variétés de groupe’, Invent. Math. 40(2) (1977), 171–193].

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 $n\equiv11\pmod{24}$.

We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.

Given an elliptic curve E in Legendre form $y^2 = x(x - 1)(x - \lambda )$ over the fraction field of a Henselian ring R of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of E over R that depends only on the valuation of $\lambda $. We provide several examples along with an easy corollary concerning $2$-torsion.

We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minimal list of twists of $X(p)$ that have to be considered, based on local information at 2 and 3; this list depends on $p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$. We solve the equation completely when $p=11$, which previously was the smallest unresolved $p$. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on $X_{0}(11)$ defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case $p=13$. The source code for the various computations is supplied as supplementary material with the online version of this article.

A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz’s $2$-variable $p$-adic $L$-functions) and algebraic objects (two ‘everywhere unramified’ Iwasawa modules) involving codimension two cycles in a $2$-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field $K$ (where an odd prime $p$ splits) of an elliptic curve $E$, defined over $\mathbb{Q}$, with good supersingular reduction at $p$. On the analytic side, we consider eight pairs of $2$-variable $p$-adic $L$-functions in this setup (four of the $2$-variable $p$-adic $L$-functions have been constructed by Loeffler and a fifth $2$-variable $p$-adic $L$-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the $\mathbb{Z}_{p}^{2}$-extension of $K$. We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.

Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion point $P\,\in \,E\left( \mathbb{Q} \right)$, there is at most one integral multiple $\left[ n \right]P$ such that $n\,>\,C$. The proof is a modification of a proof of Ingram giving an unconditional, but not uniform, bound. The new ingredient is a collection of explicit formulæ for the sequence $v\left( {{\Psi }_{n}} \right)$ of valuations of the division polynomials. For $P$ of non-singular reduction, such sequences are already well described in most cases, but for $P$ of singular reduction, we are led to define a new class of sequences called elliptic troublemaker sequences, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on $\widehat{h}\left( P \right)/h\left( E \right)$ for integer points having two large integral multiples.

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field ${{\mathbb{F}}_{p}}$, we study the frequency to which a given group $G$ arises as a group of points $E\left( {{\mathbb{F}}_{p}} \right)$. It is well known that the only permissible groups are of the form ${{G}_{m,\,k}}\,:=\,\mathbb{Z}\,/m\mathbb{Z}\,\times \,\mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M\left( {{G}_{m,\,k}} \right)$ be the frequency to which the group ${{G}_{m,\,k}}$ arises in this way. Previously, C.David and E. Smith determined an asymptotic formula for $M\left( {{G}_{m,\,k}} \right)$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M\left( {{G}_{m,\,k}} \right)$, pointwise and on average. In particular, we show that $M\left( {{G}_{m,\,k}} \right)$ is bounded above by a constant multiple of the expected quantity when $m\,\le \,{{k}^{A}}$ and that the conjectured asymptotic for $M\left( {{G}_{m,\,k}} \right)$ holds for almost all groups ${{G}_{m,\,k}}$ when $m\,\le \,{{k}^{1/4-\in }}$. We also apply our methods to study the frequency to which a given integer $N$ arises as a group order $\#E\left( {{\mathbb{F}}_{p}} \right)$.

We study the growth of $\unicode[STIX]{x0428}$ and $p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive ${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are $p$-adic and $l$-adic Lie extensions for $l\neq p$, in particular cyclotomic and other $\mathbb{Z}_{l}$-extensions.

For an elliptic curve $E/\mathbb{Q}$ without complex multiplication we study the distribution of Atkin and Elkies primes $\ell$, on average, over all good reductions of $E$ modulo primes $p$. We show that, under the generalized Riemann hypothesis, for almost all primes $p$ there are enough small Elkies primes $\ell$ to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in $(\log p)^{4+o(1)}$ expected time.

In this paper we consider ordinary elliptic curves over global function fields of characteristic $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$. We present a method for performing a descent by using powers of the Frobenius and the Verschiebung. An examination of the local images of the descent maps together with a duality theorem yields information about the global Selmer groups. Explicit models for the homogeneous spaces representing the elements of the Selmer groups are given and used to construct independent points on the elliptic curve. As an application we use descent maps to prove an upper bound for the naive height of an $S$-integral point on $A$. To illustrate our methods, a detailed example is presented.

Let be a field complete with respect to a discrete valuation whose residue field is perfect of characteristic . We prove that every smooth, projective, geometrically irreducible curve of genus one defined over with a non-zero divisor of degree a power of has a solvable point over .

We determine conductor exponent, minimal discriminant and fibre type for elliptic curves over discrete valued fields of equal characteristic 3. Along the same lines, partial results are obtained in equal characteristic 2.

Let $\overline{{{\mathbb{Q}}_{2}}}$ be an algebraic closure of ${{\mathbb{Q}}_{2}}$ and $K$ be an unramified finite extension of ${{\mathbb{Q}}_{2}}$ included in $\overline{{{\mathbb{Q}}_{2}}}$. Let $E$ be an elliptic curve defined over $K$ with additive reduction over $K$, and having an integral modular invariant. Let us denote by ${{K}_{nr}}$ the maximal unramified extension of $K$ contained in $\overline{{{\mathbb{Q}}_{2}}}$. There exists a smallest finite extension $L$ of ${{K}_{nr}}$ over which $E$ has good reduction. We determine in this paper the degree of the extension $L/{{K}_{nr}}$.

The Tate-Drinfeld module is defined by Gekeler. We define the Tate- Drinfeld map and show the analogous properties concerning Tate elliptic curves and Tate map.