The aim of the paper is to produce new families of irreducible polynomials, generalizing previous results in the area. One example of our general result is that for a near-separated polynomial, i.e., polynomials of the form $F(x,y)=f_1(x)f_2(y)-f_2(x
)f_1(y)$, then $F(x,y)+r$ is always irreducible for any constant $r$ different from zero. We also provide the biggest known family of HIP polynomials in several variables. These are polynomials $p(x_1,ldots,x_n) in K[x_1,ldots,x_n]$ over a zero characteristic field $K$ such that $p(h_1(x_1),ldots,h_n(x_n))$ is irreducible over $K$ for every $n$-tuple $h_1(x_1),ldots,h_n(x_n)$ of non constant one variable polynomials over $K$. The results can also be applied to fields of positive characteristic, with some modifications.
In this paper we apply results from the theory of congruences of modular forms (control of reducible primes, level-lowering), the modularity of elliptic curves and Q-curves, and a couple of Frey curves of Fermat-Goldbach type, to show the existence o
f newforms of weight 2 and trivial nebentypus with coefficient fields of arbitrarily large degree and square-free or almost square-free level. More precisely, we prove that for any given numbers t and B, there exists a newform f of weight 2 and trivial nebentypus whose level N is square-free (almost square-free), N has exactly t prime divisors (t odd prime divisors and a small power of 2 dividing it, respectively), and the degree of the field of coefficients of f is greater than B.
Let $E$ be an elliptic curve over $Q$. It is well known that the ring of endomorphisms of $E_p$, the reduction of $E$ modulo a prime $p$ of ordinary reduction, is an order of the quadratic imaginary field $Q(pi_p)$ generated by the Frobenius element
$pi_p$. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of $a_p^2-4p$, the discriminant of the characteristic polynomial of the Frobenius element. In this paper, we are interested in the function $pi_{E,r,h}(x)$ counting the number of primes $p$ up to $x$ such that $a_p^2-4p$ is square-free and in the congruence class $r$ modulo $h$. We give in this paper the precise asymptotic for $pi_{E,r,h}(x)$ when averaging over elliptic curves defined over the rationals, and we discuss the relation of this result with the Lang-Trotter conjecture, and with some other problems related to the curve modulo $p$.
In a paper of P. Paillier and J. Villar a conjecture is made about the malleability of an RSA modulus. In this paper we present an explicit algorithm refuting the conjecture. Concretely we can factorize an RSA modulus n using very little information
on the factorization of a concrete n coprime to n. However, we believe the conjecture might be true, when imposing some extra conditions on the auxiliary n allowed to be used. In particular, the paper shows how subtle the notion of malleability is.
We solve the diophantine equations x^4 + d y^2 = z^p for d=2 and d=3 and any prime p>349 and p>131 respectively. The method consists in generalizing the ideas applied by Frey, Ribet and Wiles in the solution of Fermats Last Theorem, and by Ellenberg
in the solution of the equation x^4 + y^2 = z^p, and we use Q-curves, modular forms and inner twists. In principle our method can be applied to solve this type of equations for other values of d.
