Average Frobenius distribution for the degree two primes of a number field

Abstract Let K be a number field and r an integer. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree two prime ideals of K with trace of Frobenius equal to r. Under certain restrictions on K, we show that “on average” the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

[1]  Ethan Smith.  A VARIANT OF THE BARBAN-DAVENPORT-HALBERSTAM THEOREM , 2011, 1201.6007.

[2]  Neil J. Calkin,et al.  Average Frobenius distributions for elliptic curves over abelian extensions , 2011 .

[3]  K. James,et al.  Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals , 2011, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  S. Baier.  The Lang-Trotter conjecture on average , 2006, math/0609095.

[5]  Glyn Harman,et al.  ANALYTIC NUMBER THEORY (American Mathematical Society Colloquium Publications 53) , 2005 .

[6]  K. James.  Average frobenius distributions for elliptic curves with 3-torsion , 2004 .

[7]  M. Ram Murty,et al.  Problems in analytic number theory , 2000 .

[8]  M. Murty,et al.  On the Distribution of Supersingular Primes , 1996, Canadian Journal of Mathematics.

[9]  René Schoof,et al.  Nonsingular plane cubic curves over finite fields , 1987, J. Comb. Theory A.

[10]  H. W. Lenstra,et al.  Factoring integers with elliptic curves , 1987 .

[11]  M. Taylor.  INTRODUCTION TO CYCLOTOMIC FIELDS(Graduate Texts in Mathematics, 83) , 1983 .

[12]  Helmut Hasse.  Number Theory , 1979 .

[13]  H. Trotter,et al.  Frobenius Distributions in GL2-Extensions: Distribution of Frobenius Automorphisms in GL2-Extensions of the Rational Numbers , 1976 .

[14]  Thomas W. Judson,et al.  Abstract Algebra , 2020, Mathemagics: A Magical Journey Through Advanced Mathematics.

[15]  M. Deuring.  Die Typen der Multiplikatorenringe elliptischer Funktionenkörper , 1941 .

[16]  D. Ivanov,et al.  Average Frobenius distributions for elliptic curves with nontrivial rational torsion , 2005 .

[17]  Francesco Pappalardi,et al.  Average Frobenius distribution for inerts in $Bbb Q(i)$ , 2004 .

[18]  F. Pappalardi.  AVERAGE FROBENIUS DISTRIBUTION FOR INERTS IN Q(i) , 2003 .

[19]  N. Elkies.  Distribution of supersingular primes , 1991 .

[20]  H. Iwaniec,et al.  Analytic Number Theory , 2021, All the Math You Missed.

[21]  H. Trotter,et al.  Frobenius Distributions in GL2-Extensions , 1976 .

[22]  D. A. Burgess.  On Character Sums and L-Series. II , 1962 .