Mathematics

Bent functions from a vector space V n over F 2 of even dimension n=2m into the cyclic group Z 2 k , or equivalently, relative difference sets in V n × Z 2 k with forbidden subgroup Z 2 k , can be obtained from spreads of V n for any k≤n/2 . In this article, existence and construction of bent functions from V n to Z 2 k , which do not come from the spread construction is investigated. A construction of bent functions from V n into Z 2 k , k≤n/6 , (and more generally, into any abelian group of order 2 k ) is obtained from partitions of F 2 m × F 2 m , which can be seen as a generalization of the Desarguesian spread. As for the spreads, the union of a certain fixed number of sets of these partitions is always the support of a Boolean bent function.

Elliptic curves arise in many important areas of modern number theory. One way to study them is take local data, the number of solutions modulo p , and create an L -function. The behavior of this global object is related to two of the seven Clay Millennial Problems: the Birch and Swinnerton-Dyer Conjecture and the Generalized Riemann Hypothesis. We study one-parameter families over Q(T) , which are of the form y 2 = x 3 +A(T)x+B(T) , with non-constant j -invariant. We define the r-th moment of an elliptic curve to be A r,E (p):= 1 p ??tmodp a t (p ) r , where a t (p) is p minus the number of solutions to y 2 = x 3 +A(t)x+B(t)modp . Rosen and Silverman showed biases in the first moment equal the rank of the Mordell-Weil group of rational solutions. Michel proved that p A 2,E (p)= p 2 +O( p 3/2 ) . Based on several special families where computations can be done in closed form, Miller in his thesis conjectured that the largest lower-order term in the second moment that does not average to 0 is on average negative. He further showed that such a negative bias has implications in the distribution of zeros of the elliptic curve L -function near the central point. To date, evidence for this conjecture is limited to special families. In addition to studying some additional families where the calculations can be done in closed form, we also systematically investigate families of various rank. These are the first general tests of the conjecture; while we cannot in general obtain closed form solutions, we discuss computations which support or contradict the conjecture. We then generalize to higher moments, and see evidence that the bias continues in the even moments.

We employ a variant of Wright's circle method to determine the bivariate asymptotic behaviour of Fourier coefficients for a wide class of eta-theta quotients with simple poles in H .

We study the number of points in the family of plane curves defined by a trinomial C(α,β)={(x,y)??F 2 q :α x a 11 y a 12 +β x a 21 y a 22 = x a 31 y a 32 } with fixed exponents (not collinear) and varying coefficients over finite fields. We prove that each of these curves has an almost predictable number of points, given by a closed formula that depends on the coefficients, exponents, and the field, with a small error term N(α,β) that is bounded in absolute value by 2 g ~ q 1/2 , where g ~ is a constant that depends only on the exponents and the field. A formula for g ~ is provided, as well as a comparison of g ~ with the genus g of the projective closure of the curve over F q ¯ ¯ ¯ ¯ ¯ . We also give several linear and quadratic identities for the numbers N(α,β) that are strong enough to prove the estimate above, and in some cases, to characterize them completely.

In this article, Eisenstein cohomology of the arithmetic group G 2 (Z) with coefficients in any finite dimensional highest weight irreducible representation has been determined. We accomplish this by studying the cohomology of the boundary of the Borel-Serre compactification.

Zeckendorf proved that every positive integer n can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this decomposition to construct a two-player game. Given a fixed integer n and an initial decomposition of n=n F 1 , the two players alternate by using moves related to the recurrence relation F n+1 = F n + F n−1 , and whoever moves last wins. The game always terminates in the Zeckendorf decomposition; depending on the choice of moves the length of the game and the winner can vary, though for n≥2 there is a non-constructive proof that Player 2 has a winning strategy. Initially the lower bound of the length of a game was order n (and known to be sharp) while the upper bound was of size nlogn . Recent work decreased the upper bound to of size n , but with a larger constant than was conjectured. We improve the upper bound and obtain the sharp bound of 5 √ +3 2 n−IZ(n)− 1+ 5 √ 2 Z(n) , which is of order n as Z(n) is the number of terms in the Zeckendorf decomposition of n and IZ(n) is the sum of indices in the Zeckendorf decomposition of n (which are at most of sizes logn and log 2 n respectively). We also introduce a greedy algorithm that realizes the upper bound, and show that the longest game on any n is achieved by applying splitting moves whenever possible.

Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the "small" solutions of these equations. Under "small" solutions we mean the solutions, say, with absolute values or sizes ??10 100 . Such algorithms were formerly constructed for Thue equations, relative Thue equations. The relative and inhomogeneous Thue equations have applications in solving index form equations and certain resultant form equations. It is also known that certain "totally real" relative Thue equations can be reduced to absolute Thue equations (equations over Z ). As a common generalization of the above results, in our paper we develop a fast algorithm for calculating "small" solutions (say with sizes ??10 100 ) of inhomogeneous relative Thue equations, more exactly of certain inequalities that generalize those equations. We shall show that in the "totally real" case these can similarly be reduced to absolute inhomogeneous Thue inequalities. We also give an application to solving certain resultant equations in the relative case.

We give an asymptotic formula for the number of weak Campana points of bounded height on a family of orbifolds associated to norm forms for Galois extensions of number fields. From this formula we derive an asymptotic for the number of elements with m -full norm over a given Galois extension of Q . We also provide an asymptotic for Campana points on these orbifolds which illustrates the vast difference between the two notions, and we compare this to the Manin-type conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado.

This is a report of the author's talk at RIMS workshop 2020 Problems and Prospects in Analytic Number Theory held online on Zoom. We discuss a recent formulation of log Manin's conjecture for klt Campana points and an approach to this conjecture using the height zeta function method.

The notion of chiral prime concatenations is studied as a recursive construction of prime numbers starting from a seed set and with appropriate blocks to define the primality growth, generation by generation, either from the right or from the left. Several basic questions are addressed like whether chiral concatenation is a symmetric process, an endless process, as well as the calculation of largest chiral prime numbers. In particular, the largest left-concatenated prime number is constructed. It is a unique prime number of 24 digits. By introducing anomalous left-concatenations of primes we can surpass the limit of 24 digits for left-concatenated primes. It is conjectured that prime numbers are left chiral under anomalous concatenations.