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Number Theory  | enregistrements trouvés : 119

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Birch gave an extremely efficient algorithm to compute a certain subspace of classical modular forms using the Hecke action on classes of ternary quadratic forms. We extend this method to compute all forms of non-square level using the spinor norm, and we exhibit an implementation that is very fast in practice. This is joint work with Jeffery Hein and Gonzalo Tornaria.

11E20 ; 11F11 ; 11F37 ; 11F27

Post-edited  Zeta functions and monodromy
Veys, Wim (Auteur de la Conférence) | CIRM (Editeur )

The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of $f$, its local monodromy. We will discuss in this survey talk rationality issues for these zeta functions and the origins of the conjecture. The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of ...

14D05 ; 11S80 ; 11S40 ; 14E18 ; 14J17

Erdös and Sárközy asked the maximum size of a subset of the first $N$ integers with no two elements adding up to a perfect square. In this talk we prove that the tight answer is $\frac{11}{32}N$ for sufficiently large $N$. We are going to prove some stability results also. This is joint work with Simao Herdade and Ayman Khalfallah.

05A18 ; 11B75

Post-edited  A refinement of the abc conjecture
Stewart, Cameron L. (Auteur de la Conférence) | CIRM (Editeur )

We shall discuss joint work with Robert and Tenenbaum on a proposed refinement of the well known abc conjecture.

11N25 ; 11Dxx ; 11N56

Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the corresponding nonlinear group of morphims of affine three space. Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the ...

11G05 ; 37A45

Spherical Hecke algebra, Satake transform, and an introduction to local Langlands correspondence.

20C08 ; 22E50 ; 11S37

We present heuristics that suggest that there is a uniform bound on the rank of $E(\mathbb{Q})$ as $E$ varies over all elliptic curves over $\mathbb{Q}$. This is joint work with Jennifer Park, John Voight, and Melanie Matchett Wood.

11R29 ; 11G40 ; 11G05 ; 14H52 ; 11R45

We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, Zhang and Maynard-Tao, respectively. We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a ...

11N05 ; 11B05

Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the number field case and on a way to strengthen it assuming a height conjecture. During the second part we will focus on function fields of positive characteristic and describe a new result obtained in a joined work with Pacheco. Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the ...

14G05 ; 11G35

We study cascades of bifurcations in a simple family of maps on the circle, and connect this behavior to the geometry of an absolute period leaf in genus $2$. The presentation includes pictures of an exotic foliation of the upper half plane, computed with the aid of the Möller-Zagier formula.

30F10 ; 30F30

Post-edited  Large gaps between primes in subsets
Maynard, James (Auteur de la Conférence) | CIRM (Editeur )

All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long strings of consecutive composite values of a polynomial. This is joint work with Ford, Konyagin, Pomerance and Tao. All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long ...

11N05 ; 11N35 ; 11N36

Post-edited  Braids and Galois groups
Matzat, B. Heinrich (Auteur de la Conférence) | CIRM (Editeur )

arithmetic fundamental group - Galois theory - braid groups - rigid analytic geometry - rigidity of finite groups

12F12 ; 11R32 ; 20F36 ; 20D08

Post-edited  Unramified graph covers of finite degree
Li, Winnie (Auteur de la Conférence) | CIRM (Editeur )

Given a finite connected undirected graph $X$, its fundamental group plays the role of the absolute Galois group of $X$. The familiar Galois theory holds in this setting. In this talk we shall discuss graph theoretical counter parts of several important theorems for number fields. Topics include
(a) Determination, up to equivalence, of unramified normal covers of $X$ of given degree,
(b) Criteria for Sunada equivalence,
(c) Chebotarev density theorem.
This is a joint work with Hau-Wen Huang.
Given a finite connected undirected graph $X$, its fundamental group plays the role of the absolute Galois group of $X$. The familiar Galois theory holds in this setting. In this talk we shall discuss graph theoretical counter parts of several important theorems for number fields. Topics include
(a) Determination, up to equivalence, of unramified normal covers of $X$ of given degree,
(b) Criteria for Sunada equivalence,
(c) Chebotarev density ...

05C25 ; 05C50 ; 11R32 ; 11R44 ; 11R45

Recently, Armstrong, Reiner and Rhoades associated with any (well generated) complex reflection group two parking spaces, and conjectured their isomorphism. This has to be seen as a generalisation of the bijection between non-crossing and non-nesting partitions, both counted by the Catalan numbers. In this talk, I will review the conjecture and discuss a new approach towards its proof, based on the geometry of the discriminant of a complex reflection group. This is an ongoing joint project with Iain Gordon. Recently, Armstrong, Reiner and Rhoades associated with any (well generated) complex reflection group two parking spaces, and conjectured their isomorphism. This has to be seen as a generalisation of the bijection between non-crossing and non-nesting partitions, both counted by the Catalan numbers. In this talk, I will review the conjecture and discuss a new approach towards its proof, based on the geometry of the discriminant of a complex ...

06B15 ; 05A19 ; 55R80

We discuss the distribution of the trace of a random matrix in the compact Lie group USp2g, with the normalized Haar measure. According to the generalized Sato-Tate conjecture, if A is an abelian variety of dimension g defined over the rationals, the sequence of traces of Frobenius in the successive reductions of A modulo primes appears to be equidistributed with respect to this distribution. If g = 2, we provide expressions for the characteristic function, the density, and the repartition function of this distribution in terms of higher transcendental functions, namely Legendre and Meijer functions. We discuss the distribution of the trace of a random matrix in the compact Lie group USp2g, with the normalized Haar measure. According to the generalized Sato-Tate conjecture, if A is an abelian variety of dimension g defined over the rationals, the sequence of traces of Frobenius in the successive reductions of A modulo primes appears to be equidistributed with respect to this distribution. If g = 2, we provide expressions for the cha...

11G05 ; 11G10 ; 14G10 ; 37C30

The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on convergence results and what can be said about the limits.

37A05 ; 37A25 ; 37A15

We give a survey on recent advances in Grothendiek's program of anabelian geometry to characterize arithmetic and geometric objects in Galois theoretic terms. Valuation theory plays a key role in these developments, thus confirming its well deserved place in mainstream mathematics.
The talk notes are available in the PDF file at the bottom of the page.

12F10 ; 12J10 ; 12L12

Post-edited  The category MF in the semistable case
Faltings, Gerd (Auteur de la Conférence) | CIRM (Editeur )

For smooth schemes the category $MF$ (defined by Fontaine for DVR's) realises the "mysterious functor", and provides natural systems of coeffients for crystalline cohomology. We generalise it to schemes with semistable singularities. The new technical features consist mainly of different methods in commutative algebra

14F30

Post-edited  On Schmidt's subspace theorem
Evertse, Jan-Hendrik (Auteur de la Conférence) | CIRM (Editeur )

Last year, I published together with Roberto Ferretti a new version of the quantitative subspace theorem, giving a better upper bound for the number of subspaces containing the solutions of the system of inequalities involved. In my lecture, I would like to discuss this improvement, and go into some aspects of its proof.

11J13 ; 11J68

Post-edited  Taming the coloured multizetas
Ecalle, Jean (Auteur de la Conférence) | CIRM (Editeur )

1. We shall briefly describe the ARI-GARI structure; recall its double origin in Analysis and mould theory; explain what makes it so well-suited to the study of multizetas; and review the most salient results it led to, beginning with the exchanger $adari(pal^\bullet)$ of double symmetries $(\underline{al}/\underline{il}) \leftrightarrow (\underline{al}/\underline{al})$, and culminating in the explicit decomposition of multizetas into a remarkable system of irreducibles, positioned exactly half-way between the two classical multizeta encodings, symmetral resp. symmetrel.

2. Although the coloured, esp. two-coloured, multizetas are in many ways more regular and better-behaved than the plain sort, their sheer numbers soon make them computationally intractable as the total weight $\sum s_i$ increases. But help is at hand: we shall show a conceptual way round this difficulty; make explicit its algebraic implementation; and sketch some of the consequences.

A few corrections and comments about this talk are available in the PDF file at the bottom of the page.
1. We shall briefly describe the ARI-GARI structure; recall its double origin in Analysis and mould theory; explain what makes it so well-suited to the study of multizetas; and review the most salient results it led to, beginning with the exchanger $adari(pal^\bullet)$ of double symmetries $(\underline{al}/\underline{il}) \leftrightarrow (\underline{al}/\underline{al})$, and culminating in the explicit decomposition of multizetas into a ...

11M32

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