THE ERROR TERM IN THE SATO–TATE THEOREM OF BIRCH

2019/02/08 by M. Ram Murty, M. RAM MURTY, Neha Prabhu +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Calculus (dental) #Elliptic curve #Measure (data warehouse) #Term (time) #Type (biology) #math.NT #msc:11G05 #msc:11K38

paper · pdf · doi:10.1017/s0004972718001648

openalex publication_date 2019/02/08 · openalex created_date 2019/02/21 · arxiv created 2019/06/08 · arxiv updated 2019/06/11 · openalex updated_date 2026/08/05

Abstract

We establish an error term in the Sato–Tate theorem of Birch. That is, for p prime, q=p^r and an elliptic curve E:y²=x³+ax+b , we show that \begineqnarray#\(a,b)∈ \mathbbF_q²:\unicode[STIX]x1D703_a,b∈ I\=\unicode[STIX]x1D707_ST(I)q²+O_r(q^7/4)\endeqnarray for any interval I⊆ [0,\unicode[STIX]x1D70B] , where the quantity \unicode[STIX]x1D703_a,b is defined by 2√(q)cos \unicode[STIX]x1D703_a,b=q+1-E(\mathbbF_q) and \unicode[STIX]x1D707_ST(I) denotes the Sato–Tate measure of the interval I .

Citations

• Factoring Integers with Elliptic Curves
• La conjecture de Weil : I
• Die Typen der Multiplikatorenringe elliptischer Funktionenkörper
• La conjecture de Weil : II