2021/03/23 by Dohoon Choi, Choi, Dohoon
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2103.12533
openalex publication_date 2021/03/23 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Let F be a totally real field, and mathbbAF be the adele ring of F.\nLet us fix N to be a positive integer. Let \π1=\⊗\π1,v and\n\π2=\⊗\π2,v be distinct cohomological cuspidal automorphic\nrepresentations of \GLn( mathbbAF) with levels less than or\nequal to N.\n Let \N(\π1,\π2) be the minimum of the absolute norm of v\n nmid \∞ such that \π1,v not \≃ \π2,v and that \π1,v\nand \π2,v are unramified. We prove that there exists a constant CN\nsuch that for every pair \π1 and \π2,
mathcalN(
pi1,
pi2)
leq\nCN. This improves known bounds
mathcalN(
pi1,
pi2)=O(QA)
;
;
;\n(
textsome A
text depending only on n), where Q is the maximum of\nthe analytic conductors of \π1 and \π2.\n This result applies to newforms on \Γ1(N). In particular, assume that\nf1 and f2 are Hecke eigenforms of weight k1 and k2 on\n\SL2(\ℤ), respectively. We prove that if for all p \∈\n 2,7 ,
lambdaf1(p)/
sqrtp(k1-1) =\n
lambdaf2(p)/
sqrtp(k2-1), then f1=cf2 for some constant c.\nHere, for each prime p, \λfi(p) denotes the p-th Hecke\neigenvalue of fi.\n