The Lang–Trotter conjecture for products of non-CM elliptic curves
Tóm tắt
Inspired by the work of Lang–Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over
$${\mathbb {Q}}$$
and by the subsequent generalization of Cojocaru–Davis–Silverberg–Stange to generic abelian varieties, we study the analogous question for abelian surfaces isogenous to products of non-CM elliptic curves over
$${\mathbb {Q}}$$
that are not
$${\overline{{\mathbb {Q}}}}$$
-isogenous. We formulate the corresponding conjectural asymptotic, provide upper bounds, and explicitly compute (when the elliptic curves lie outside a thin set) the arithmetically significant constants appearing in the asymptotic. This allows us to provide computational evidence for the conjecture.
Tài liệu tham khảo
Akbary, A., Parks, J.: On the Lang-Trotter conjecture for two elliptic curves. Ramanujan J. 49(3), 585–623 (2019). https://doi.org/10.1007/s11139-018-0050-7
Baier, S., Patankar, V.M.: Applications of the square sieve to a conjecture of Lang and Trotter for a pair of elliptic curves over the rationals. In: Geometry, Algebra, Number Theory, and Their Information Technology Applications. vol. 251. Springer Proc. Math. Stat. Springer, Cham, pp. 39–57 (2018). https://doi.org/10.1007/978-3-319-97379-1_3
Baier, S., Jones, N.: A refined version of the Lang-Trotter conjecture. Int. Math. Res. Not. 2009 (2008). https://doi.org/10.1093/imrn/rnn136
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24.3-4 (1997). Computational Algebra and Number Theory (London, 1993), pp. 235–265. https://doi.org/10.1006/jsco.1996.0125
Cojocaru, A.C., Davis, R., Silverberg, A., Stange, K.E.: Arithmetic properties of the Frobenius traces defined by a rational abelian variety (with two appendices by J-P. Serre). Int. Math. Res. Not. IMRN 12, 3557–3602 (2017). https://doi.org/10.1093/imrn/rnw058
Cojocaru, A.-C., Grant, D., Jones, N.: One-parameter families of elliptic curves over Q with maximal Galois representations. Proc. Lond. Math. Soc. 103(4), 654–675 (2011). https://doi.org/10.1112/plms/pdr001
Cojocaru, A.C., Jones, N., Serban, V., Wang, T.: Bounds for distributions of Frobenius traces on abelian varieties with small Sato-Tate group (in preparation)
Daniels, H.B.: An infinite family of Serre curves. J. Number Theory 155, 226–247 (2015). https://doi.org/10.1016/j.jnt.2015.03.016
Daniels, H.B., Hatley, J., Ricci, J.: Elliptic curves with maximally disjoint division fields. Acta Arith. 175(3), 211–223 (2016)
David, C., Pappalardi, F.: Average Frobenius distributions of elliptic curves. Int. Math. Res. Not. 4, 165–183 (1999). https://doi.org/10.1155/S1073792899000082
Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hansischen Univ. 14, 197–272 (1941). https://doi.org/10.1007/BF02940746
Fité, F., Kedlaya, K.S., Rotger, V., Sutherland, A.V.: Sato-Tate distributions and Galois endomorphism modules in genus 2. Compos. Math. 148(5), 1390–1442 (2012). https://doi.org/10.1112/S0010437X12000279
Fouvry, E., Murty, M.R.: On the distribution of supersingular primes. Can. J. Math. 48(1), 81–104 (1996). https://doi.org/10.4153/CJM-1996-004-7
Harris, M.: Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications. In: Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, vol. II. vol. 270. Progr. Math. Birkhäuser Boston, Inc., Boston, pp. 1–21 (2009). https://doi.org/10.1007/978-0-8176-4747-6_1
Jones, N.: Averages of elliptic curve constants. Math. Ann. 345(3), 685–710 (2009). https://doi.org/10.1007/s00208-009-0373-1
Jones, N.: Almost all elliptic curves are Serre curves. Trans. Am. Math. Soc. 362(3), 1547–1570 (2010). https://doi.org/10.1090/S0002-9947-09-04804-1
Jones, N.: Pairs of elliptic curves with maximal Galois representations. J. Number Theory 133(10), 3381–3393 (2013). https://doi.org/10.1016/j.jnt.2013.03.002
Katz, N.M.: Lang-Trotter revisited. Bull. Am. Math. Soc. 46(3), 413–457 (2009). https://doi.org/10.1090/S0273-0979-09-01257-9
Lagarias, J.C., Odlyzko, A.M.: Effective versions of the Chebotarev density theorem. In: Algebraic Number Fields: L-Functions and Galois Properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 409–464 (1977)
Lang, S., Trotter, H.: Frobenius Distributions in GL2-Extensions. Lecture Notes in Mathematics, vol. 504. Springer, Berlin (1976)
Murty, V.K.: Frobenius distributions and Galois representations. In: Automorphic Forms, Automorphic Representations, and Arithmetic (Fort Worth, TX, 1996), vol. 66. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, pp. 193–211 (1999)
Oesterlé, J.: Réduction modulo \(p^{n}\) des sous-ensembles analytiques fermés de \(\mathbf{Z}^{N}_{p}\). Invent. Math. 66(2), 325–341 (1982). https://doi.org/10.1007/BF01389398
Ribet, K.: On \(\ell \)-adic representations attached to modular forms. Invent. Math. 28, 245–275 (1975). https://doi.org/10.1007/bf01425561
Rubin, K., Silverberg, A.: Mod 2 representations of elliptic curves. Proc. Am. Math. Soc. 129, 53 (1999). https://doi.org/10.2307/2669028
Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4), 259–331 (1972)
Serre, J.-P.: Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Etudes Sci. Publ. Math. 54, 323–401 (1981)
Silverberg, A.: Explicit families of elliptic curves with prescribed mod N representations. In: Modular Forms and Fermat’s Last Theorem (Boston, MA, 1995), pp. 447–461. Springer, New York (1997)
The LMFDB Collaboration. The L-Functions and Modular Forms Database (2013). http://www.lmfdb.org. Accessed 16 Sept 2013
The Sage Developers. SageMath, the Sage Mathematics Software System (Version 9.0) (2020). https://www.sagemath.org