Mathematics

Given a gonality γ and a prime power q?�γ�?1 we show that for every large genus g there exists a curve C defined over F q of genus g and gonality γ and with exactly γ(q+1) F q -rational points. This is the maximal number of rational points allowed. This answers a recent conjecture by Faber--Grantham. Our methods are based on Poonen's work on squarefree values of polynomials together with a Newton polygon argument.

We construct curves of each genus g≥2 for which Coleman's effective Chabauty bound is sharp and Coleman's theorem can be applied to determine rational points if the rank condition is satisfied. We give numerous examples of genus two and rank one curves for which Coleman's bound is sharp. Based on one of those curves, we construct an example of a curve of genus five whose rational points are determined using the descent method together with Coleman's theorem.

Under special conditions, we prove that the set of preperiodic points for semigroups of self-morphisms of affine spaces falling on cyclotomic closures is not dense. generalising results of Ostafe and Young (2020). We also extend previous results about boundness of house and height on certain preperiodicity sets of higher dimension in semigroup dynamics.

Let D be the ring of S -integers in a global field and D ^ its profinite completion. We discuss the relation between density in D and the Haar measure of D ^ : in particular, we ask when the density of a subset X of D is equal to the Haar measure of its closure in D ^ . In order to have a precise statement, we give a general definition of density which encompasses the most commonly used ones. Using it we provide a necessary and sufficient condition for the equality between density and measure which subsumes a criterion due to Poonen and Stoll. In another direction, we extend the Davenport-Erdős theorem to every D as above and offer a new interpretation of it as a "density=measure" result. Our point of view also provides a simple proof that in any D the set of elements divisible by at most k distinct primes has density 0 for any natural number k . Finally, we show that the group of units of D ^ is contained in the closure of the set of irreducible elements of D .

In 2013, Strauch asked how various sequences of real numbers defined from trigonometric functions such as x n =(cosn ) n distributed themselves (mod1) . Strauch's inquiry is motivated by several such distribution results. For instance, Luca proved that the sequence x n =(cosαn ) n (mod1) is dense in [0,1] for any fixed real number α such that α/π is irrational. Here we generalise Luca's results to other sequences of the form x n =f(n ) n (mod1) . We also examine the size of the set |{n≤N:r<|cos(nπα) | n }| where 0<r<1 and α are fixed such that α/π is irrational.

We make two algorithms that generate all prime numbers up to a given limit, they are a development of sieve of Eratosthenes algorithm, we use two formulas to achieve this development, where all the multiples of prime number 2 are eliminated in the first formula, and all the multiples of prime numbers 2 and 3 are eliminated in the second formula. Using the first algorithm we proof sieve of Sundaram's algorithm, then we improve it to be more efficient prime generating algorithm. We will show the difference in performance between all the algorithms we will make and sieve of Eratosthenes algorithm in terms of run time.

Let F q be a finite field with q= p n elements. In this paper, we study the number of solutions of equations of the form a 1 x d 1 1 +?? a s x d s s =b with x i ??F p t i , where b??F q and t i |n for all i=1,??s . In our main results, we employ results on quadratic forms to give an explicit formula for the number of solutions of diagonal equations with restricted solution sets satisfying certain natural restrictions on the exponents. As a consequence, we present conditions for the existence of solutions. We also discuss further questions concerning equations with restricted solution sets and present some open problems.

Let V(k) be projective variety over a number field k and let K be a finite extension of k . We classify the K -isomorphisms of V(k) in terms of the Serre C ∗ -algebra of V(k) . As an application, a new proof of the Faltings Finiteness Theorem for the rational points on the higher genus curves is suggested.

A Diophantine m -tuple over K is a set of m non-zero (distinct) elements of K with the property that the product of any two distinct elements is one less than a square in K . Let X:( x 2 ??)( y 2 ??)( z 2 ??)= k 2 , be a threefold. Its K -rational points parametrize Diophantine triples over K such that the product of the elements of the triple that corresponds to the point (x,y,z,k)?�X(K) is equal to k . We denote by X ¯ ¯ ¯ ¯ the projective closure of X and for a fixed k by X k a fibre over k . First, we prove that the variety X ¯ ¯ ¯ ¯ is rational which leads us to a new rational parametrization of the set of Diophantine triples. Next, specializing to finite fields, we find a Shioda-Inose structure of the K3 surface X k for a given k??F ? p in the prime field F p of odd characteristic, determined by an abelian surface which is a square E k ? E k of an explicit elliptic curve. We derive an explicit formula for N(p,k) , the number of Diophantine triples over F p with the product of elements equal to k . Moreover, we show that the variety X ¯ ¯ ¯ ¯ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on X ¯ ¯ ¯ ¯ over an arbitrary finite field F q . We reprove the formula for the number of Diophantine triples over F q from Dujella-Kazalicki(2021). From the interplay of the two (K3 and rational) fibrations of X ¯ ¯ ¯ ¯ , we derive the formula for the second moment of the elliptic surface E k (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating S 4 ( ? 0 (8)) . Finally, in the Appendix, Luka Lasi? defines circular Diophantine m -tuples and describes the parametrization of these sets.

We study p -adic L -functions L p (s,?) for Dirichlet characters ? . We show that L p (s,?) has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of ? . The expansion is proved by transforming a known formula for p -adic L -functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the p -adic Dirichlet series. We also provide an alternative proof of the expansion using p -adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for c=2 , where we obtain a Dirichlet series expansion that is similar to the complex case.