Related papers: On Types of Elliptic Pseudoprimes

We establish supercongruences for two kinds of Ap\'ery-like numbers, which involve Bernoulli numbers and Bernoulli polynomials. Conjectural supercongruences of the same type for another four kinds of Ap\'ery-like numbers are also proposed.

Let $L/k$ be a finite abelian extension of an imaginary quadratic number field $k$. Let $\mathfrak{p}$ denote a prime ideal of $\mathcal{O}_k$ lying over the rational prime $p$. We assume that $\mathfrak{p}$ splits completely in $L/k$ and…

We construct new rational approximants of Euler's constant that improve those of Aptekarev et al. (2007) and Rivoal (2009). The approximants are given in terms of certain (mixed type) multiple orthogonal polynomials associated with the…

In this note, we consider an l-isogeny descent on a pair of elliptic curves over Q. We assume that l > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finite- dimensional…

In this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the "weak main…

Fix a prime number $p$. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and $p$, which also contains the primitive $p$-th root of unity $\mu_p$. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin,…

Let A be an abelian variety defined over a number field F. For a prime number $\ell$, we consider the field extension of F generated by the $\ell$-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we…

We construct uncountably categorical 3-nilpotent groups of exponent p > 3. They are not one-based and do not allow the interpretation of an infinite field. Therefore they are counterexamples to Zilbers Conjecture. First 2-nilpotent new…

After a short review of the historical milestones on normal numbers, we introduce the Borel numbers as the reals admitting a probability function on their different bases representations. In this setting, we provide two probabilistic…

Let $E/\mathbb{Q}$ be an elliptic curve and let $p\ge 5$ be a prime of good supersingular reduction. We generalize results due to Meng Fai Lim proving Kida's formula and integrality results for characteristic elements of signed Selmer…

In this paper, we study umbral calculus to have alternative ways of obtaining our results. That is, we derive some interesting identities of the higher-order Bernoulli, Euler and Hermite polynomials arising from umbral calculus to have…

For an elliptic curve, we study how many Selmer groups are cotorsion over the anti-cyclotomic $\mathbb{Z}_p$-extension as one varies the prime $p$ or the quadratic imaginary field in question.

The complexity of the elliptic curve method of factorization (ECM) is proven under the celebrated conjecture of existence of smooth numbers in short intervals. In this work we tackle a different version of ECM which is actually much more…

Given a closed symplectic manifold, we construct invariants which count (a) closed rational pseudoholomorphic curves with prescribed cusp singularities and (b) punctured rational pseudoholomorphic curves with ellipsoidal negative ends. We…

Recently the new q-Euler numbers are defined. In this paper we derive the the Kummer type congruence related to q-Euler numbers and we introduce some interesting formulae related to these q-Euler numbers.

New exceptional (i.e. non-repeating) prime number multiplets are given and formulated in terms of arithmetic progressions, along with laws governing them. Accompanying repeating prime number multiplets are pointed out. Prime number…

Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field $F$ with good supersingular reduction at all primes above $p$. We study both the classical and plus/minus Selmer groups over the cyclotomic…

We study the complexity (that is, the weight of the multiplication table) of the elliptic normal bases introduced by Couveignes and Lercier. We give an upper bound on the complexity of these elliptic normal bases, and we analyze the weight…

A Hermitian-symplectic metric is a Hermitian metric whose K\"ahler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be…

This paper contains a survey of supersingular isogeny graphs associated to supersingular elliptic curves and their various applications to cryptography. Within limitation of space, we attempt to address a broad audience and make this part…