Mathematics

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal $\fl$ of $\F_q[T]$, the question essentially asks whether, up to isogeny, a Drinfeld module ϕ over $\F_q(T)$ contains a rational $\fl$-torsion point if the reduction of ϕ at almost all primes of $\F_q[T]$ contains a rational $\fl$-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2 , but negative if the rank is 3 . Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

Suppose K is N -dimensional local field of characteristic p , G=Gal( K sep /K) , G <p is the maximal quotient of G of period p and nilpotent class <p and K <p ⊂ K sep is such that Gal( K <p /K)= G <p . We use nilpotent Artin-Schreier theory to identify G <p with the group G(L) obtained from a profinite Lie F p -algebra L via the Campbell-Hausdorff composition law. The canonical P -topology on K is used to define a dense Lie subalgebra L P in L . The algebra L P can be provided with a system of P -topological generators and its P -open subalgebras correspond to all N -dimensional extensions of K in K <p . These results are applied to higher local fields K of characteristic 0 containing primitive p -th root of unity. If Γ=Gal( K alg /K) we introduce similarly the quotient Γ <p =G(L) , a dense F p -Lie algebra L P ⊂L , and describe the structure of L P in terms of generators and relations. The general result is illustrated by explicit presentation of Γ <p modulo third commutators.

Arithmetic functions in Number Theory meet the Sprague-Grundy function from Combinatorial Game Theory. We study a variety of 2-player games induced by standard arithmetic functions, such as Euclidian division, divisors, remainders and relatively prime numbers, and their negations.

Our aim is to find a complex continued fraction algorithm finding all the best Diophantine approximations to a complex number. Using the sequence of minimal vectors in a two dimensional lattice over Gaussian integers, we obtain an algorithm defined on a submanifold of the space of unimodular two dimensional Gauss lattices. This submanifold is transverse to the diagonal flow. Thanks to the correspondence between minimal vectors and best Diophantine approximations, the algorithm finds all the best approximations to a complex number. A byproduct of the algorithm is the best constant for the complex version of Dirichlet Theorem about approximations of complex numbers by quotients of Gaussian integers.

This paper extends the concept of a B -happy number, for B?? , from the rational integers, Z , to the Gaussian integers, Z[i] . We investigate the fixed points and cycles of the Gaussian B -happy functions, determining them for small values of B and providing a method for computing them for any B?? . We discuss heights of Gaussian B -happy numbers, proving results concerning the smallest Gaussian B -happy numbers of certain heights. Finally, we prove conditions for the existence and non-existence of arbitrarily long arithmetic sequences of Gaussian B -happy numbers.

Let p be a prime number, F a totally real number field unramified at places above p and D a quaternion algebra of center F split at places above p and at no more than one infinite place. Let v be a fixed place of F above p and r ¯ ¯ :Gal( F ¯ ¯ ¯ ¯ /F)→ GL 2 ( F ¯ ¯ ¯ p ) an irreducible modular continuous Galois representation which, at the place v , is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of GL 2 ( F v ) over F ¯ ¯ ¯ p associated to r ¯ ¯ in the corresponding Hecke-eigenspaces of the mod p cohomology have Gelfand--Kirillov dimension [ F v :Q] , as well as several related results.

In a recent work, the present author generalized a fundamental result of Gauss related to quadratic reciprocity, and also showed that the above result of Gauss is equivalent to a special case of a well known result of Sylvester related to the Frobenius coin problem. In this note, we show that the above generalization of the result of Gauss naturally leads to an interesting generalization of the result of Sylvester. To be precise, for given coprime positive integers a and b , and for a family of values of k in the interval 0?�k??a??)(b??) , we find the number of natural numbers ?�k , which can be expressed in the form ax+by for non-negative integers x and y . Further, we find a special case of the generalization of Gauss's result that is equivalent to the general version of Sylvester's result. This result naturally leads us to another proof of Gauss's Lemma and Eisenstein's Lemma for Jacobi symbols.

In this article, we prove that a general version of Alladi's formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom A or Axiom A # . As applications, we apply our main results to certain semigroups coming from algebraic number theory, arithmetical geometry and graph theory, particularly generalizing the results of Wang 2021, Kural et al. 2020 and Duan et al. 2020.

For non-negative integers l 1 , l 2 ,?? l n , we define character sums ? ( l 1 , l 2 ,?? l n ) and ? ( l 1 , l 2 ,?? l n ) over a finite field which are generalizations of Jacobsthal and modified Jacobsthal sums, respectively. We express these character sums in terms of Greene's finite field hypergeometric series. We then express the number of points on the hyperelliptic curves y 2 =( x m +a)( x m +b)( x m +c) and y 2 =x( x m +a)( x m +b)( x m +c) over a finite field in terms of the character sums ? ( l 1 , l 2 , l 3 ) and ? ( l 1 , l 2 , l 3 ) , and finally obtain expressions in terms of the finite field hypergeometric series.

In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over Q . We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1 . In addition, we show that these families of quadratic twists satisfy the 2 -part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.