Final Report Summary - EQUIARITH (Equidistribution in number theory)
One of the most venerable problems in mathematics (going back at least to the greeks) is to study the solution of polynomial equations with integral coordinates; an example is the problem of representing a given integer as a sum of three squares. A solution to this problem (a triple of integers whose sum of squares is the given number) is called a represention. Determining the structure of the set of such representations is a fascinating problem whose resolution involves a blend of surprisingly number of difficult methods from number theory, harmonic analysis and ergodic theory.
One approach goes back to the late 50’s and the work of the Russian Mathematician Y. V. Linnik and is called “ergodic method”. Although this was recognized only very recently, this method is, in a way, an ancestor –in the arithmetic context- of methods which have developed 10 years later (the theory of geodesic and more generally Anosov flows). Another approach, due to H. Iwaniec and W. Duke, goes back to the late 80s and is very different and builds on methods from more analytic techniques building on the theory of automorphic forms and their associated L-functions.
The main achievement of the present project has been to reconcile these two very different approaches and to combine them to make new progress on related longstanding problems. The key ingredient to pass from one to the other is the notion of “automorphic period”. Periods are fundamental objects in modern number theory and have multiple incarnations (arithmetic: they are related to sets of solutions of some integral polynomial equations; analytic: periods occur as special values of L-function (like the Riemann zeta function) ; geometric: they occur as integrals along orbits for some group action). Therefore, the understanding of periods from one viewpoint bring new lights on the others: this is the strategy that has been followed by the PI Ph. Michel, the three key team members of the project, M. Einsiedler, E. Lindenstrauss, and A. Venkatesh and their collaborators.
A first example is the complete resolution of the so-called “subconvexity problem” ( for GL(1) and GL(2) L-functions) by Ph. M. and A.V. The subconvexity problem is a important question in the analytic theory of L-functions which is closely related to the celebrated Riemann hypothesis. The solution which is a culmination of a century of progresses uses precisely the relation between periods and L-functions, and has been strongly inspired by ergodic techniques.
A second example is the work of the four key team members who combined the resolution of the subconvexity problem with cutting-edge ergodic theoretic techniques developed separately by M. Einsiedler and E. Lindenstrauss (together with A. Katok) to settle a 40 years old conjecture of Y. V. Linnik, on the distribution properties of integral matrices satisfying some polynomial equation (Linnik's conjecture was a higher dimensional version of the problem of studying the representations of integers as a sum of three squares).
A third example is the uncovering (jointly with E. Kowalski and E. Fouvry) of a new type of equidistribution results which combines group theory, periods and their automorphic forms with algebraic geometric methods. It has been realized recently that such results shed new lights on many classical question from analytic number theory (like the study of the prime numbers).
Besides these results, the research developed during this project contributed to increase significantly our understanding of the rich relationships between number theory and ergodic theory and help to make further progress in both fields.
One approach goes back to the late 50’s and the work of the Russian Mathematician Y. V. Linnik and is called “ergodic method”. Although this was recognized only very recently, this method is, in a way, an ancestor –in the arithmetic context- of methods which have developed 10 years later (the theory of geodesic and more generally Anosov flows). Another approach, due to H. Iwaniec and W. Duke, goes back to the late 80s and is very different and builds on methods from more analytic techniques building on the theory of automorphic forms and their associated L-functions.
The main achievement of the present project has been to reconcile these two very different approaches and to combine them to make new progress on related longstanding problems. The key ingredient to pass from one to the other is the notion of “automorphic period”. Periods are fundamental objects in modern number theory and have multiple incarnations (arithmetic: they are related to sets of solutions of some integral polynomial equations; analytic: periods occur as special values of L-function (like the Riemann zeta function) ; geometric: they occur as integrals along orbits for some group action). Therefore, the understanding of periods from one viewpoint bring new lights on the others: this is the strategy that has been followed by the PI Ph. Michel, the three key team members of the project, M. Einsiedler, E. Lindenstrauss, and A. Venkatesh and their collaborators.
A first example is the complete resolution of the so-called “subconvexity problem” ( for GL(1) and GL(2) L-functions) by Ph. M. and A.V. The subconvexity problem is a important question in the analytic theory of L-functions which is closely related to the celebrated Riemann hypothesis. The solution which is a culmination of a century of progresses uses precisely the relation between periods and L-functions, and has been strongly inspired by ergodic techniques.
A second example is the work of the four key team members who combined the resolution of the subconvexity problem with cutting-edge ergodic theoretic techniques developed separately by M. Einsiedler and E. Lindenstrauss (together with A. Katok) to settle a 40 years old conjecture of Y. V. Linnik, on the distribution properties of integral matrices satisfying some polynomial equation (Linnik's conjecture was a higher dimensional version of the problem of studying the representations of integers as a sum of three squares).
A third example is the uncovering (jointly with E. Kowalski and E. Fouvry) of a new type of equidistribution results which combines group theory, periods and their automorphic forms with algebraic geometric methods. It has been realized recently that such results shed new lights on many classical question from analytic number theory (like the study of the prime numbers).
Besides these results, the research developed during this project contributed to increase significantly our understanding of the rich relationships between number theory and ergodic theory and help to make further progress in both fields.