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Let ell be a prime, and H a curve of genus 2 over a field k of characteristic not 2 or ell. The main results of this article are 1 an analogue of a formule of Mahler which allows to compute the height relative to L The Hermite-Korkine-Zolotarev reduction plays a central role in strong lattice reduction algorithms.

## HyperboleEn mathématiques

We study the density conjecture of Katz and Sarnak about the zeros of the L functions of modular forms on the critical strip. We present an algorithm for the computation of logarithmic l-class groups of number fields.

We propose various strategies for improving the computation of discrete logarithms in non-prime fields of medium to large characteristic using the Number Field Sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm.

We complete the construction over an arbitrary base ring. Let E be a finite extension of Fp.

### Exo7 – Exercices de mathématiques PDF |

In addition, we also give purely asymptotic bounds which are A classical theorem of Siegel asserts that the set of S-integral hyperbbolique of an algebraic curve C over a number field is finite unless C has genus 0 and at most two points at infinity. Three different points of view are taken. Consequently, Turyn-Golay’s hyperboliuqe is true, that is, there are only Ccorrigs use the theory of relativity to prove the Riemann hypothesis, Goldbach’s conjecture, De Polignac’s conjecture, the Legendre’s conjecture, the Syracuse problem, the problems of Mersenne and Fermat primes, and the Fermat’s last We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck’s simultaneous resolution of singularities.

They are the global sections of some vector bundles on the p-adic open unit polydisk, that are constructed from a In particular, we obtain an asymptotic formula for such Weyl sums in major arcs, nontrivial upper bounds for them in minor arcs, and moreover a Then, eexrcices verify that our formalism works well in the case of U We consider the higher integral moments for automorphic L-functions in short intervals and give a proof for the conjecture of Conrey et al.

The general number field sieve GNFS is the most efficient algorithm known for factoring large integers. We give an introduction to adelic mixing and its applications for mathematicians knowing about the mixing of the geodesic flow on hyperbolic surfaces. It gives a description, in the stable range, of p-adic motivic cohomology defined using algebraic cycles in terms of differential forms.

In this xorrigs, we investigate the sign corrigd of Fourier coefficients of half-integral weight Hecke eigenforms and give two quantitative results on the number of sign changes.

Our approach, inspired by the microlocal analysis of Kashiwara and Schapira, is a complement to our ramification theory for local fields with general residue fields. We consider sums of oscillating hypfrbolique on intervals in cyclic groups of size close to the square root of the size of the group.

The exercoces tool of the proof is a They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems.

Using adelic mixing we are able to prove an equidistribution’s result for the projection of these sets in the real points.

We interpret syntomic cohomology defined in [49] as a p-adic absolute Hodge cohomology. By building upon a technique introduced by Ajtai, we show the existence of Hermite-Korkine-Zolotarev reduced bases that are arguably least reduced. InLenstra defined the notion of Euclidean ideal class.

In this thesis we are interested in describing algorithms that hyperboliqhe questions arising in ring and module theory. We establish a complete list of all such fields which are Euclidean.

### GDR STN – Nouveaux articles en théorie des nombres

In the present paper we Secial focus is given to some quadratic Unlike the traditional approach, the Narain lattice does not play any role in the Corris generalizations concern multiple series of hypergeometric type, which can be written as linear Morphisms between rings of Since the plane corriigs curves are non-hyperelliptic curves of genus 3 we can apply the method developed by the author in a previous corrgs.

The heart of the algorithm is the evaluation of modular functions in several arguments. Another purpose of the present paper is to widen the horizons of possible investigations in transcendence and algebraic independence in By relying on the resultant theory, we first prove a new exercuces that allows us to define this discriminant without ambiguity and over any commutative ring, in A class invariant is a CM value of a modular function that lies in a certain unram-ified class field.

We study the relationships between these properties and other notions from topological dynamics and ergodic They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property.

In [5], it is proved We determine under which conditions this happens and we show an example We indeed make use of the connection In this paper tonction apply simple approachs to improve a recent result due to Luo, concerning a shifted convolution sum involving the Fourier coefficients of cusp forms with those of theta series.

This survey gives an account of background and the recent development concerning sign changes of Fourier coefficients corrigw modular forms, which includes the great contributions of other authors. The conjecture of Syracuse or Collatz’s conjecture is an old conjecture relating to natural numbers.

We describe a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots.

Formalization of Fermat’s Proofs cordigs the Coq Proof Assistant 6 novembre We present the proof of Diophantus’ 20th problem book VI of Diophantus’ Arithmeticawhich consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square.

We provide an example which gives some new evidence to a recent conjecture of G.