2006/02/09 by François Brunault, Francois Brunault, Brunault, Francois
Mathematics · #11F67 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F67
paper · pdf · doi:10.48550/arxiv.math/0602186
155 pages, PhD thesis, French, with an appendix by Loic Merel
arxiv created 2006/02/09 · openalex publication_date 2006/02/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the special value at 2 of L-functions of modular forms of weight 2 on congruence subgroups of the modular group. We prove an explicit version of Beilinson's theorem for the modular curve X1(N). When N is prime, we deduce that the target space of Beilinson's regulator map is generated by the images of Milnor symbols associated to modular units of X1(N). We also suggest a reformulation of Zagier's conjecture on L(E,2) for the jacobian J1(N) of X1(N), where E is an elliptic curve of conductor N. In this direction we define an analogue of the elliptic dilogarithm for any jacobian J : it is a function RJ from the complex points of J to a finite-dimensional vector space. In the case J=J1(N), we establish a link between the aforementioned L-values and the function RJ evaluated at \Q-rational points of the cuspidal subgroup of J.