Serre curves relative to obstructions modulo 2

We consider elliptic curves $E / \mathbb{Q}$ for which the image of the adelic Galois representation $\rho_E$ is as large as possible given a constraint on the image modulo 2. For such curves, we give a characterization in terms of their $\ell$-adic images, compute all examples of conductor at most 500,000, precisely describe the image of $\rho_E$, and offer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves

On a Classification of 3 adic Galois images associated to isogeny torsion graphs

Let $E$ be a non CM elliptic curve defined over \Q. There is an isogeny torsion graph associated to $E$ and there is also a Galois representation \rho_{E,l^{\infty}} \colon \Gal(\Qbar/\Q) \to \GL_2(\ZZ_{\ell}) associated to $E$ for every prime $\ell.$ In this article, we explore relation between these two objects when $\ell=3$. More precisely, we give a classification of 3 adic Galois images associated to vertices of isogeny torsion graph of $E.

On the metaphysics of F1\mathbb F_1

In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of $\mathbb S[\mu_{n,+}]$-polynomials and relate this concept to the known notion of number systems. The Riemann-Roch theorem for the ring $\mathbb Z$ of the integers that we obtained recently uses the understanding of $\mathbb Z$ as a ring of polynomials $\mathbb S[X]$ in one variable over the absolute base $\mathbb S$, where $1+1=X+X^2$. The absolute base $\mathbb S$ (the categorical version of the sphere spectrum) thus turns out to be a strong candidate for the incarnation of the mysterious $\mathbb F_1$.Comment: Dedicated to Yuri Manin, 14 Figure

The Average Number OF Divisors in Certain Arithmetic Sequences

In this paper we study the sum p≤xτ(np), where τ(n) denotes the number of divisors of n, and {np} is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is x as x → ∞. As an application, we consider the case where the sequence is given by the Fourier coefficients of a modular form

A Like ELGAMAL Cryptosystem But Resistant To Post-Quantum Attacks

The Modulo 1 Factoring Problem (M1FP) is an elegant mathematical problem which could be exploited to design safe cryptographic protocols and encryption schemes that resist to post quantum attacks. The ELGAMAL encryption scheme is a well-known and efficient public key algorithm designed by Taher ELGAMAL from discrete logarithm problem. It is always highly used in Internet security and many other applications after a large number of years. However, the imminent arrival of quantum computing threatens the security of ELGAMAL cryptosystem and impose to cryptologists to prepare a resilient algorithm to quantum computer-based attacks. In this paper we will present a like-ELGAMAL cryptosystem based on the M1FP NP-hard problem. This encryption scheme is very simple but efficient and supposed to be resistant to post quantum attacks