Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Factoring newparts of Jacobians of certain modular curves

Author(s): M. Ram Murty; Kaneenika Sinha
Journal: Proc. Amer. Math. Soc. 138 (2010), 3481-3494.
MSC (2010): Primary 11F11, 11F25, 11F30, 11G10, 11G18
Posted: May 5, 2010
MathSciNet review: 2661548
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We prove a conjecture of Yamauchi which states that the level for which the new part of 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 and level 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

References:

1.: A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma_0(m)$ , Math. Ann., Vol. 185 (1970), 134-160. MR 0268123 (42:3022)
2.: M. H. Baker, E. González-Jiménez, J. González and B. Poonen, Finiteness results for modular curves of genus at least , Amer. J. Math., Vol. 127 (2005), 1325-1387. MR 2183527 (2006i:11065)
3.: D. Bressoud and S. Wagon, A course in computational number theory, Key College Publishing, Emeryville, CA, 2000. MR 1756372 (2001f:11200)
4.: H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, 138, Springer, 1993. MR 1228206 (94i:11105)
5.: H. Cohen, Sur les tels que soit $\mathbb{Q}$ -Isogène à un produit de courbes elliptiques, preprint, available on http://www.ufr-mi.u-bordeaux.fr/ $\sim$ cohen
6.: L. E. Dickson, History of the theory of numbers, Vol. 3, Chelsea Publishing Co., New York, 1966. MR 0245499 (39:6807a)
7.: M. Eichler, On the class number of imaginary quadratic fields and the sums of divisors of natural numbers, J. Indian Math. Soc., Vol. 19 (1955), 153-180. MR 0080769 (18:299a)
8.: E. Halberstadt and A. Kraus, Courbes de Fermat: résultats et problèmes, J. Reine Angew. Math., Vol. 548 (2002), 167-234. MR 1915212 (2003h:11068)
9.: C. Hamer, A formula for the traces of the Hecke operators on certain spaces of newforms, Arch. Math. (Basel), Vol. 70 (1998), no. 3, 204-210. MR 1604056 (99b:11045)
10.: M. N. Huxley, A note on polynomial congruences, Recent Progress in Analytic Number Theory, Vol. 1 (Durham, 1979), Academic Press, London-New York, 1981, 193-196. MR 637347 (83e:10005)
11.: A. Knightly and C. Li, Traces of Hecke operators, Mathematical Surveys and Monographs, 133, American Mathematical Society, Providence, RI, 2006. MR 2273356 (2008g:11090)
12.: 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.
13.: G. Martin, Dimensions of the spaces of cusp forms and newforms on $\Gamma_0(N)$ and $\Gamma_1(N)$ , J. Number Theory, Vol. 112 (2005), no. 2, 197-240. MR 2141534 (2005m:11069)
14.: M. Ram Murty, Problems in analytic number theory, Graduate Texts in Mathematics, 206, second edition, Springer, 2007. MR 2376618 (2008j:11001)
15.: M. Ram Murty and K. Sinha, Effective equidistribution of eigenvalues of Hecke operators, J. Number Theory, Vol. 129 (2009), no. 3, 681-714. MR 2488597
16.: S. Ramanujan, Highly composite numbers, Collected Papers of Srinivasa Ramanujan, AMS Chelsea Publ., Amer. Math. Soc., Providence, RI, 2000, 78-128. MR 2280858
17.: K. A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann., Vol. 253 (1980), no. 1, 43-62. MR 594532 (82e:10043)
18.: J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$ , Math. Comp., Vol. 29 (1975), 243-269. MR 0457373 (56:15581a)
19.: E. Royer, Facteurs $\mathbb{Q}$ -simples de $J\sb 0(N)$ de grande dimension et de grand rang, Bull. Soc. Math. France., Vol. 128 (2000), no. 2, 219-248. MR 1772442 (2001j:11041)
20.: J.-P. Serre, Répartition asymptotique des valeurs propres de l'opérateur de Hecke , J. Amer. Math. Soc., Vol. 10 (1997), no. 1, 75-102. MR 1396897 (97h:11048)
21.: G. Shimura, On the factors of the Jacobian variety of a modular function field, J. Math. Soc. Japan, Vol. 25 (1973), 523-544. MR 0318162 (47:6709)
22.: T. Yamauchi, On $\mathbb{Q}$ -simple factors of Jacobian varieties of modular curves, Yokohama Math. J., Vol. 53 (2007), no. 2, 149-160. MR 2302608 (2008k:11062)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|