Geometric and Analytic Number Theory

• Applicants: Professor Dr. Valentin Blomer; Professor Dr. Jörg Brüdern; Professor Dr. Ulrich Derenthal
• Subject Area: Mathematics
• Term: from 2014 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 255083470

• Final Report Year: 2024

Final Report Abstract

The central goal of the project “Geometric and analytic number theory” was to develop and apply geometric and analytic techniques to solve number theoretic problems. A considerable number of publications in high level journals document substantial progress in a variety of aspects of which we mention a representative selection: The analysis of multi-dimensional character and exponential sums, based on deep results in algebraic geometry, was applied to a variety of problems in number theory, in particular to analytic properties of L-functions. Using a combination of analytic and algebraic arguments, new cases of Manin’s conjecture were established which asks for the asymptotic distribution of rational points on algebraic varieties. For integral points, results regarding their distribution, the Hasse principle, Brauer–Manin obstructions and and arithmetic purity of strong approximation were obtained. The phenomenon of spectral reciprocity formulae for L-functions was observed, systematically developed and applied, among other things, to subconvexity results for L-functions. The first instance of a higher rank version of Sarnak’s density conjecture was established which provides an interpolation bound for the number of possible exceptional eigenvalues on arithmetic locally symmetric spaces.

Publications

• Arithmetic purity of strong approximation for semi-simple simply connected groups. Compositio Mathematica, 156(12), 2628-2649.
  Cao, Yang & Huang, Zhizhong
  
• The split torsor method for Manin’s conjecture. Transactions of the American Mathematical Society, 373(12), 8485-8524.
  Derenthal, Ulrich & Pieropan, Marta
  
• A symplectic restriction problem. Mathematische Annalen, 382(3-4), 1323-1424.
  Blomer, Valentin & Corbett, Andrew
  
• Uniform Titchmarsh divisor problems. Advances in Mathematics, 393(2021, 12), 108076.
  Assing, Edgar; Blomer, Valentin & Li, Junxian
  
• Density theorems for GL(n). Inventiones mathematicae, 232(2), 683-711.
  Blomer, Valentin
  
• INTEGRAL POINTS ON SINGULAR DEL PEZZO SURFACES. Journal of the Institute of Mathematics of Jussieu, 23(3), 1259-1294.
  Derenthal, Ulrich & Wilsch, Florian
  
• On Waring’s problem for larger powers. Journal für die reine und angewandte Mathematik relles Journal), 0(0).
  Brüdern, Jörg & Wooley, Trevor D.
  
• Simultaneous equidistribution of toric periods and fractional moments of $L$-functions. Journal of the European Mathematical Society, 26(8), 2745-2796.
  Blomer, Valentin & Brumley, Farrell
  
• The Weyl bound for triple product L-functions. Duke Mathematical Journal, 172(6).
  Blomer, Valentin; Jana, Subhajit & Nelson, Paul D.
  
• The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one. Forum of Mathematics, Sigma, 12(2024).
  Blomer, Valentin; Brüdern, Jörg; Derenthal, Ulrich & Gagliardi, Giuliano