

Contemporary Problems in Number Theory
June 2, 2016 13:00, Moscow, Steklov Mathematical Institute, Room 530 (8 Gubkina)






padic analyticity, powerful and smooth moduli and strong subconvexity
Djordje Milicevic^{ab} ^{a} Max Planck Institute for Mathematics
^{b} Bryn Mawr College

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Abstract:
Divisor problem, distribution of primes in arithmetic progressions, or, say, equidistribution of rational points on varieties  all cornerstone problems in analytic number theory  can see their complexity rise in several directions, including the "level aspect", in which the modulus is increasing. Within it, the socalled "depth" and "smooth" aspects, where the modulus is highly powerful or wellfactorable, respectively, play a distinctive role, with tools often paralleling those available in the archimedean direction. In this talk, we will discuss several manifestations of this phenomenon.
In particular, we will present our recent subconvexity bound for the central value of a Dirichlet Lfunction of a character to a prime power modulus, which breaks a longstanding barrier known as the Weyl exponent. To this end, we develop a general method to estimate short exponential sums involving padically analytic fluctuations, which can be naturally seen as a padic analogue of the method of exponent pairs. Natural analogues of the circle method and large sievetype inequalities and their consequences for subconvexity and moments of Lfunctions (joint work with Blomer) will also be mentioned.
Language: English

