Factoring newparts of Jacobians of certain modular curves

Murty, M.; Sinha, Kaneenika

doi:10.1090/S0002-9939-10-10376-1

Factoring newparts of Jacobians of certain modular curves
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by M. Ram Murty and Kaneenika Sinha PDF

Proc. Amer. Math. Soc. 138 (2010), 3481-3494 Request permission

Abstract:

We prove a conjecture of Yamauchi which states that the level $N$ for which the new part of $J_0(N)$ is $\mathbb {Q}$-isogenous to a product of elliptic curves is bounded. We also state and partially prove a higher-dimensional analogue of Yamauchi’s conjecture. In order to prove the above results, we derive a formula for the trace of Hecke operators acting on spaces $S^{new}(N,k)$ of newforms of weight $k$ and level $N.$ We use this trace formula to study the equidistribution of eigenvalues of Hecke operators on these spaces. For any $d\geq 1,$ we estimate the number of normalized newforms of fixed weight and level, whose Fourier coefficients generate a number field of degree less than or equal to $d.$

References

A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134–160. MR 268123, DOI 10.1007/BF01359701
Matthew H. Baker, Enrique González-Jiménez, Josep González, and Bjorn Poonen, Finiteness results for modular curves of genus at least 2, Amer. J. Math. 127 (2005), no. 6, 1325–1387. MR 2183527, DOI 10.1353/ajm.2005.0037
David Bressoud and Stan Wagon, A course in computational number theory, Key College Publishing, Emeryville, CA; in cooperation with Springer-Verlag, New York, 2000. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 1756372
Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
H. Cohen, Sur les $N$ tels que $J_0(N)$ soit $\mathbb {Q}$-Isogène à un produit de courbes elliptiques, preprint, available on http://www.ufr-mi.u-bordeaux.fr/$\sim$cohen
Leonard Eugene Dickson, History of the theory of numbers. Vol. I: Divisibility and primality. , Chelsea Publishing Co., New York, 1966. MR 0245499
Martin Eichler, On the class of imaginary quadratic fields and the sums of divisors of natural numbers, J. Indian Math. Soc. (N.S.) 19 (1955), 153–180 (1956). MR 80769
Emmanuel Halberstadt and Alain Kraus, Courbes de Fermat: résultats et problèmes, J. Reine Angew. Math. 548 (2002), 167–234 (French, with English summary). MR 1915212, DOI 10.1515/crll.2002.058
Carol Hamer, A formula for the traces of the Hecke operators on certain spaces of newforms, Arch. Math. (Basel) 70 (1998), no. 3, 204–210. MR 1604056, DOI 10.1007/s000130050185
M. N. Huxley, A note on polynomial congruences, Recent progress in analytic number theory, Vol. 1 (Durham, 1979) Academic Press, London-New York, 1981, pp. 193–196. MR 637347
Andrew Knightly and Charles Li, Traces of Hecke operators, Mathematical Surveys and Monographs, vol. 133, American Mathematical Society, Providence, RI, 2006. MR 2273356, DOI 10.1090/surv/133
C.-W. Lim, Decomposition of $S_k(\Gamma _0(N))$ over $\mathbb {Q}$ and variants of partial nim, PhD thesis, University of California at Berkeley, 2005.
Greg Martin, Dimensions of the spaces of cusp forms and newforms on $\Gamma _0(N)$ and $\Gamma _1(N)$, J. Number Theory 112 (2005), no. 2, 298–331. MR 2141534, DOI 10.1016/j.jnt.2004.10.009
M. Ram Murty, Problems in analytic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 206, Springer, New York, 2008. Readings in Mathematics. MR 2376618
M. Ram Murty and Kaneenika Sinha, Effective equidistribution of eigenvalues of Hecke operators, J. Number Theory 129 (2009), no. 3, 681–714. MR 2488597, DOI 10.1016/j.jnt.2008.10.010
S. Ramanujan, Highly composite numbers [Proc. London Math. Soc. (2) 14 (1915), 347–409], Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, pp. 78–128. MR 2280858
Kenneth A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), no. 1, 43–62. MR 594532, DOI 10.1007/BF01457819
J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$, Math. Comp. 29 (1975), 243–269. MR 457373, DOI 10.1090/S0025-5718-1975-0457373-7
Emmanuel Royer, Facteurs $\mathbf Q$-simples de $J_0(N)$ de grande dimension et de grand rang, Bull. Soc. Math. France 128 (2000), no. 2, 219–248 (French, with English and French summaries). MR 1772442, DOI 10.24033/bsmf.2369
Jean-Pierre Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke $T_p$, J. Amer. Math. Soc. 10 (1997), no. 1, 75–102 (French). MR 1396897, DOI 10.1090/S0894-0347-97-00220-8
Goro Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523–544. MR 318162, DOI 10.2969/jmsj/02530523
Takuya Yamauchi, On $\Bbb Q$-simple factors of Jacobian varieties of modular curves, Yokohama Math. J. 53 (2007), no. 2, 149–160. MR 2302608