A K3 surface over a number field has infinitely many rational points over a finite field extension. For K3 surfaces of degree 2, arising as double covers of P2 branched along a smooth sextic curve, we give a bound for the degree of such an extension. Moreover, using ideas of van Luijk and a surface constructed by Elsenhans and Jahnel, we give an explicit family of K3 surfaces of degree 2 defined over Q with geometric Picard number 1 and infinitely many Q-rational points that is Zariski dense in the moduli space of K3 surfaces of degree 2.

Let E be an elliptic curve defined over Q. We classify all groups that can arise as E(Q(ζp))tors up to isomorphism for any prime p. When p−1 is not divisible by small integers such as 3,4,5,7, or 11, we obtain a sharper classification. For any abelian number field K, the torsion subgroup E(K)tors is a subgroup of E(Qab)tors. Our methods provide tools to eliminate non-realized torsion structures from the list of possibilities for E(K)tors.

Let p≥3 be a prime number. A Fermat curve over Q of exponent p is defined by an equation of the form axp+byp+czp=0, where a, b, c are non-zero rational numbers. We prove in this article that there exist infinitely many Fermat curves defined over Q, of exponent p, pairwise non Q-isomorphic, contradicting the Hasse principle.

Morton and Vivaldi defined the polynomials whose roots are parabolic parameters for a one-parameter family of polynomial maps (called delta factors here). They conjectured that delta factors are irreducible for the family z↦z2+c. One can easily show the irreducibility for periods 1 and 2 by reducing it to the irreducibility of cyclotomic polynomials. However, for periods 3 and more, this becomes a challenging problem. We prove the irreducibility of delta factors for period 3 and demonstrate the existence of infinitely many irreducible delta factors for periods greater than 3.

We develop, through a governing field, genus theory for a number field K with tame ramification in T and splitting in S, where T and S are finite disjoint sets of primes of K. This approach extends the one initiated by the second author in the case of the class group. We are able to express the S-T genus number of a cyclic extension L/K of degree p in terms of the rank of a matrix constructed from the Frobenius elements of the primes ramified in L/K, in the Galois group of the underlying governing extension. For quadratic extensions L/Q, the matrices in question are constructed from the Legendre symbols of the primes ramified in L/Q and the primes of S.

For an integer n>1, let P+(n) be the largest prime factor of n. Following a celebrated conjecture of Erdős and Turán in the 1930s, Erdős and Pomerance proved in 1978 that lim infx→∞|{n≤x:P+(n+1)>P+(n)}|x>0. In this article, their result is extended to lim infx→∞|{n≤x:P+(n+1)>P+(n),μ2(n)=μ2(n+1)=1}|x>0.

We study the growth of sumsets A+B⊂S⊂G, where S does not contain an arithmetic progression of length 2k+1, and where G is a commutative group, in which every nonzero element has an order of at least 2k+1. More specifically, we show the following: if A,B⊂G are sets such that A+B does not contain an arithmetic progression of length 2k+1, then |A+B|≥|A|2k−13k−2|B|k3k−2. As an application we derive upper bounds on the cardinality of the summands in sumsets A+B+C contained in the set of t-th powers, where t≥2 is an integer. In particular, we show that min(|A|,|B|,|C|)≪(logN)4/5 for t=2, and min(|A|,|B|,|C|)≪t(logN)1/2 for t≥3.

The Shimura lift of a Hecke eigenform multiplied by a theta series is the square of the form. We extend this result by generalizing the product map to the Rankin–Cohen bracket. We prove that the Shimura lift of the Rankin–Cohen bracket of an eigenform and a theta series is given by the Rankin–Cohen bracket of the eigenform and itself.

By the circle method, an asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski-dense subset of the biprojective hypersurface $$\sum _{i=1}^nx_iy_i^2=0$$ in $\mathbb {P}^{n-1}\tim