Mathematics

Given m,n,q?�N such that q is a prime power and m?? , a??F q , we establish a sufficient condition for the existence of primitive pair (α,f(α)) in F q m such that α is normal over F q and Tr F q m / F q ( α ?? )=a , where f(x)??F q m (x) is a rational function of degree sum n . Further, when n=2 and q= 5 k for some k?�N , such a pair definitely exists for all (q,m) apart from at most 20 choices.

We study the minimal number of existential quantifiers needed to define a diophantine set over a field and relate this number to the essential dimension of the functor of points associated to such a definition.

The main point of the paper is to take the explicit motivic Chabauty-Kim method developed in papers of Dan-Cohen--Wewers and Dan-Cohen and the author and make it work for non-rational curves. In particular, we calculate the abstract form of an element of the Chabauty-Kim ideal for Z[1/?�] -points on a punctured elliptic curve, and lay some groundwork for certain kinds of higher genus curves. For this purpose, we develop an "explicit Tannakian Chabauty-Kim method" using Q p -Tannakian categories of Galois representations in place of Q -linear motives. In future work, we intend to use this method to explicitly apply the Chabauty-Kim method to a curve of positive genus in a situation where Quadratic Chabauty does not apply.

Given a number field extension K/k with an intermediate field K + fixed by a central element of the corresponding Galois group of prime order p , we build an algebraic torus over k whose rational points are elements of K × sent to k × via the norm map N K/ K + . The goal is to compute the Tamagawa number of that torus explicitly via Ono's formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when K/k is Galois. Partial results including the numerator are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra. We also present tools developed in SAGE for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. Particular attention is given to the case when [K: K + ]=2 and K is a CM-field. This case corresponds to tori in GSp 2n , and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.

When travelling from the number fields theory to the function fields theory, one cannot miss the deep analogy between rank 1 Drinfeld modules and the group of root of unity and the analogy between rank 2 Drinfeld modules and elliptic curves. But so far, there is no known structure in number fields theory that is analogous to the Drinfeld modules of higher rank r > 2. In this paper we investigate the classes of those Drinfeld modules of higher rank r > 2. We describe explicitly the Weil polynomials defining the isogeny classes of rank r Drinfeld modules for any rank r > 2. our explicit description of the Weil polynomials depends heavily on Yu's classification of isogeny classes (analogue of Honda-Tate at abelian varieties). Actually Yu has also explicitly did that work for r = 2. To complete the classification, we define the new notion of fine isomorphy invariants for any rank r Drinfeld module and we prove that the fine isomorphy invariants together with J-invariants completely determine the L-isomorphism classes of rank r Drinfeld modules defined over the finite field L.

Under GRH, we establish a version of Duke's short-sum theorem for entire Artin L -functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerical constants in this work are explicit.

Let K be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes p for which there exists an elliptic curve over K admitting a K -rational p -isogeny. This builds on work of David, Larson-Vaintrob, and Momose. Combining this algorithm with work of Bruin-Najman, ?zman-Siksek, and most recently Box, we determine the above set of primes for the four quadratic fields Q( 7 ????) , Q( ?? ????????) , Q( ??0 ??????????) , and Q( 5 ????) , providing the first such examples after Mazur's 1978 determination for K=Q . The termination of the algorithm relies on the Generalised Riemann Hypothesis.

We prove the Oppenheim conjecture for indefinite ternary diagonal forms of the type x 2 + y 2 ??β 2 z 2 where β is an irrational number. Our method is explicit in the sense that we are able to construct a solution to the problem and we obtain an effective bound on the solution. The method is geometrical and is based on elementary metric diophantine approximation.

Given a genus 2 curve C with a rational Weierstrass point defined over a number field, we construct a family of genus 5 curves that realize descent by maximal unramified abelian two-covers of C . Each of the genus 5 curves is equipped with a natural map to a genus 1 curve over an extension of the base field, which yields a model of the isogeny class of the Jacobian via restriction of scalars. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or Sage. We use these algorithms to provably compute the set of rational points of some genus 2 curves over Q of Mordell-Weil rank 2 or 3 (conditional on GRH in some cases).

Let N(?,T) denote the number of nontrivial zeros of the Riemann zeta function with real part greater than ? and imaginary part between 0 and T . We provide explicit upper bounds for N(?,T) commonly referred to as a zero density result. In 1937, Ingham showed the following asymptotic result N(?,T)=O( T 8 3 (1?��? (logT ) 5 ) . Ramaré recently proved an explicit version of this estimate. We discuss a generalization of the method used in these two results which yields an explicit bound of a similar shape while also improving the constants.