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A Titchmarsh divisor problem for elliptic curves

Published online by Cambridge University Press:  10 November 2015

PAUL POLLACK
PAUL POLLACK*
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia, U.S.A. e-mail: [email protected]
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Abstract

Let E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that

$$\begin{equation} \sum_{\substack{p \leq x \\p \equiv 1\pmod{4}}} \tau(\#E({\bf{F}}_p)) \sim \left(\frac{5\pi}{16} \prod_{p > 2} \frac{p^4-\chi(p)}{p^2(p^2-1)}\right)x, \quad\text{as $x\to\infty$}. \end{equation}$$
Here χ is the nontrivial Dirichlet character modulo 4. The proof uses number field analogues of the Brun–Titchmarsh and Bombieri–Vinogradov theorems, along with a theorem of Wirsing on mean values of nonnegative multiplicative functions.

Now suppose that E/Q is a non-CM elliptic curve. We conjecture that the sum of τ(#E(Fp)), taken over p ⩽ x of good reduction, is ~cEx for some cE > 0, and we give a heuristic argument suggesting the precise value of cE. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that this sum is ≍Ex. The proof uses combinatorial ideas of Erdős.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 1 , January 2016 , pp. 167 - 189
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[1]Akbary, A. and Felix, A. T.On invariants of elliptic curves on average. Acta Arith. 168 (2015), 31–70.CrossRefGoogle Scholar
[2]Akbary, A. and Ghioca, D.A geometric variant of Titchmarsh divisor problem. Int. J. Number Theory 8 (2012), 53–69.CrossRefGoogle Scholar
[3]Clark, P. L., Cook, B. and Stankewicz, J.Torsion points on elliptic curves with complex multiplication (with an appendix by Alex Rice). Int. J. Number Theory 9 (2013), 447–479.CrossRefGoogle Scholar
[4]Castryck, W. and Hubrechts, H.The distribution of the number of points modulo an integer on elliptic curves over finite fields. Ramanujan J. 30 (2013), 223–242.CrossRefGoogle Scholar
[5]Childress, N.Class Field Theory (Springer, New York, 2009).CrossRefGoogle Scholar
[6]Cojocaru, A. C.Reductions of an elliptic curve with almost prime orders. Acta Arith. 119 (2005), 265–289.CrossRefGoogle Scholar
[7]Duke, W.Almost all reductions modulo p of an elliptic curve have a large exponent. C. R. Math. Acad. Sci. Paris 337 (2003), 689–692.CrossRefGoogle Scholar
[8]David, C. and Wu, J.Almost prime values of the order of elliptic curves over finite fields. Forum Math. 24 (2012), 99–119.CrossRefGoogle Scholar
[9]David, C. and Wu, J.Pseudoprime reductions of elliptic curves. Canad. J. Math. 64 (2012), 81–101.CrossRefGoogle Scholar
[10]Elkies, N. D.Distribution of supersingular primes. Astérisque (Proceedings of Journées Arithmétiques, Luminy, 1989) 198–200 (1992), 127–132.Google Scholar
[11]Elliott, P. D. T. A.On the mean value of f(p). Proc. London Math. Soc. (3) 21 (1970), 28–96.CrossRefGoogle Scholar
[12]Erdős, P.On the sum ∑xk = 1d(f(k)). J. London Math. Soc. 27 (1952), 7–15.Google Scholar
[13]Friedlander, J. B. and Iwaniec, H.Divisor weighted sums. Zap. Nauchn. Sem. POMI 322 (2005), 212–219.Google Scholar
[14]Felix, A. T. and Ram Murty, M.On the asymptotics for invariants of elliptic curves modulo p. J. Ramanujan Math. Soc. 28 (2013), 271–298.Google Scholar
[15]Gekeler, E.-U.The distribution of group structures on elliptic curves over finite prime fields. Doc. Math. 11 (2006), 119–142 (electronic).CrossRefGoogle Scholar
[16]Gun, S. and Ram Murty, M.Divisors of Fourier coefficients of modular forms. New York J. Math. 20 (2014), 229–239.Google Scholar
[17]Granville, A.Smooth numbers: computational number theory and beyond. Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. Math. Sci. Res. Inst. Publ. 44 (Cambridge University Press, Cambridge, 2008), 267–323.Google Scholar
[18]Hinz, J. and Lodemann, M.On Siegel zeros of Hecke-Landau zeta-functions. Monatsh. Math. 118 (1994), 231–248.CrossRefGoogle Scholar
[19]Howe, E. W.On the group orders of elliptic curves over finite fields. Compositio Math. 85 (1993), 229–247.Google Scholar
[20]Halberstam, H. and Richert, H.-E.Sieve methods. London Math. Soc. Monogr. 4 (Academic Press, London-New York, 1974).Google Scholar
[21]Huxley, M. N.The large sieve inequality for algebraic number fields. III. Zero-density results. J. London Math. Soc. (2) 3 (1971), 233–240.CrossRefGoogle Scholar
[22]Hardy, G. H. and Wright, E. M.An introduction to the theory of numbers (sixth ed.) (Oxford University Press, Oxford, 2008).CrossRefGoogle Scholar
[23]Jiménez Urroz, J.Almost prime orders of CM elliptic curves modulo p. Algorithmic Number Theory. Lecture Notes in Comput. Sci. 5011 (Springer, Berlin, 2008), 74–87.CrossRefGoogle Scholar
[24]Lang, S.Elliptic Functions (second ed.). Grad. Texts in Math. 112 (Springer-Verlag, New York, 1987).CrossRefGoogle Scholar
[25]Lenstra, H. W. Jr.Factoring integers with elliptic curves. Ann. of Math. (2) 126 (1987), 649–673.CrossRefGoogle Scholar
[26]Linnik, U. V.The dispersion method in binary additive problems. Trans. math. monogr. 4 (Amer. Math. Soc., Providence, RI, 1963).Google Scholar
[27]Moree, P. and Cazaran, J.On a claim of Ramanujan in his first letter to Hardy. Expo. Math. 17 (1999), 289–311.Google Scholar
[28]Rajwade, A. R.Arithmetic on curves with complex multiplication by √ − 2. Proc. Camb. Phil. Soc. 64 (1968), 659–672.CrossRefGoogle Scholar
[29]Serre, J.-P.Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Invent. Math. 15 (1972), 259–331.CrossRefGoogle Scholar
[30]Shiu, P.A Brun–Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313 (1980), 161–170.Google Scholar
[31]Silverberg, A.Group order formulas for reductions of CM elliptic curves. Arithmetic, Geometry, Cryptography and Coding Theory 2009. Contemp. Math. 521 (Amer. Math. Soc., Providence, RI, 2010), 107–120.CrossRefGoogle Scholar
[32]Tao, T. Erdős's divisor bound. Blog post published July 23, 2011 at http://terrytao.wordpress.com/2011/07/23/erdos-divisor-bound/. To appear in the forthcoming book Spending Symmetry.Google Scholar
[33]Titchmarsh, E. C.A divisor problem. Rend. Circ. Mat. Palermo 54 (1930), 414–429. Errata in 57 (1933), 478–479.CrossRefGoogle Scholar
[34]Wirsing, E.Das asymptotische Verhalten von Summen über multiplikative Funktionen. Math. Ann. 143 (1961), 75–102.CrossRefGoogle Scholar
[35]Wolke, D.Multiplikative Funktionen auf schnell wachsenden Folgen. J. Reine Angew. Math. 251 (1971), 54–67.Google Scholar