The workshop brought together people attacking key modern problems in number theory via techniques involving concrete or computable descriptions. Here, was interpreted broadly, including algebraic and analytic theory, Galois arithmetic of varieties, zeta L-functions their special values, modular forms functions.

We present a very fast algorithm to build up tables of cubic fields. Real fields with discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="10 Superscript 11"> <mml:semantics> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>11</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">10^{11}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and complex down...

We prove that van Hoeij's original algorithm to factor univariate polynomials over the rationals runs in polynomial time, as well natural variants. In particular, our approach also yields time complexity results for bivariate a finite field.

This note presents our implementation in the PARI/GP system of various arithmetic invariants attached to logarithmic classes and units number fields. Our algorithms simplify improve on works Diaz y Diaz, Pauli, Pohst, Soriano second author.

We describe practical algorithms from computational algebraic number theory, with applications to class field theory. These include basic arithmetic, approximation and uniformizers, discrete logarithms computation of fields. All have been implemented in the Pari/Gp system.

We obtain the first known power-saving remainder terms for theorems of Davenport and Heilbronn on density discriminants cubic fields mean number 3-torsion elements in class groups quadratic fields. In addition, we prove analogous error quartic 2-torsion These results analytic continuation related Dirichlet series to left line R(s)=1.

Assuming the Generalized Riemann Hypothesis, Bach has shown that ideal class group $\mathcal {C}\ell _{K}$ of a number field $K$ can be generated by prime ideals having norm smaller than $12\big (\log |\mathrm {Discriminant}(K)|\big )^2$. This result is essential for computation and units Buchmann’s algorithm, currently fastest known. However, once _K$ been computed, one notices this bound could have replaced much value, so work saved. We introduce here short algorithm which allows us to...

Davenport and Heilbronn defined a bijection between classes of binary cubic forms fields, which has been used to tabulate the latter. We give simpler proof their theorem then analyze improve table-building algorithm. It computes multiplicities <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo>...

These notes contain extended abstracts on the topic of explicit methods in number theory. The range topics includes Sato-Tate conjecure, Langlands programme, function fields, L-functions and many other topics.

Nous calculons la densité des discriminants corps sextiques galoisiens de groupe S 3 , démontrant un nouveau cas conjecture Malle ainsi qu'un particulier sa généralisation par Ellenberg et Venkatesh. Plus généralement, nous étudions cubiques dans une progression arithmétique, avec zone d'uniformité plus large possible. We compute the density of Galois sextic fields with group thereby proving a new case Malle's as well special its generalization by and Further, we study cubic in an arithmetic...

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. The asymptotics of the number geodesics in starting from and returning to given cusp, horoballs parabolic fixed points universal cover , are studied this paper. case SL(2, ℤ), Bianchi groups, is developed.

Let K be a global field and f in K[X] polynomial. We present an efficient algorithm which factors polynomial time.

Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate residue of Dedekind zeta function a number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by clever use splitting primes alttext="p greater-than upper X"> <mml:mrow> <mml:mi>p</mml:mi>...

The Cohen–Lenstra–Martinet heuristics give precise predictions about the class groups of a ’random‘ number field. 3-rank quadratic fields is one few instances where these have been proven. We prove that, in this case, rate convergence at least sub-exponential. In addition, we show that defect appearing Scholz‘s mirror theorem equidistributed with respect to twisted Cohen–Lenstra density.

Call h 3 * (Δ) the number of cube roots unity in class group ℚ(Δ), where Δ is a fundamental discriminant. Davenport and Heilbronn computed mean value these numbers when tends to ±∞. The author gives general geometric argument yielding an explicit bound for error term, with additional possibility restricting arithmetic progressions. Sieve techniques then produce results about 3-parts groups Cl (ℚ(Δ)), P k almost-prime order k. In this way, one controls simultaneously both 2-rank 3-rank...