The basis problem revisited

Martin, Kimball

doi:10.1090/tran/8077

The basis problem revisited
HTML articles powered by AMS MathViewer

by Kimball Martin PDF

Trans. Amer. Math. Soc. 373 (2020), 4523-4559 Request permission

Abstract:

Eichler investigated when there is a basis of a space of modular forms consisting of theta series attached to quaternion algebras, and treated squarefree level. Hijikata, Pizer, and Shemanske completed the solution to Eichler’s basis problem for elliptic modular forms of arbitrary level by tour-de-force trace calculations. We revisit the basis problem using the representation-theoretic perspective of the Jacquet–Langlands correspondence.

Our results include: (i) a simpler proof of the solution to the basis problem for elliptic modular forms, which also allows for more flexibility in the choice of quaternion algebra; (ii) a solution to the basis problem for Hilbert modular forms; (iii) a theory of (local and global) new and old forms for quaternion algebras; and (iv) an explicit version of the Jacquet–Langlands correspondence at the level of modular forms, which is a refinement of the Hijikata–Pizer–Shemanske solution to the basis problem. Both (i) and (ii) have practical applications to computing elliptic and Hilbert modular forms. Moreover, (iii) and (iv) are desired for arithmetic applications—to illustrate, we give a simple application to Eisenstein congruences in level $p^3$.

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
Colin J. Bushnell and Guy Henniart, The local Langlands conjecture for $\rm GL(2)$, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Springer-Verlag, Berlin, 2006. MR 2234120, DOI 10.1007/3-540-31511-X
H. Carayol, Représentations cuspidales du groupe linéaire, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 2, 191–225 (French). MR 760676, DOI 10.24033/asens.1470
William Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314. MR 337789, DOI 10.1007/BF01428197
Samit Dasgupta, Henri Darmon, and Robert Pollack, Hilbert modular forms and the Gross-Stark conjecture, Ann. of Math. (2) 174 (2011), no. 1, 439–484. MR 2811604, DOI 10.4007/annals.2011.174.1.12
Lassina Dembélé and John Voight, Explicit methods for Hilbert modular forms, Elliptic curves, Hilbert modular forms and Galois deformations, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Basel, 2013, pp. 135–198. MR 3184337, DOI 10.1007/978-3-0348-0618-3_{4}
Michel Duflo and Jean-Pierre Labesse, Sur la formule des traces de Selberg, Ann. Sci. École Norm. Sup. (4) 4 (1971), 193–284. MR 437462, DOI 10.24033/asens.1210
Martin Eichler, Über die Darstellbarkeit von Modulformen durch Thetareihen, J. Reine Angew. Math. 195 (1955), 156–171 (1956) (German). MR 80768, DOI 10.1515/crll.1955.195.156
M. Eichler, The basis problem for modular forms and the traces of the Hecke operators, Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 320, Springer, Berlin, 1973, pp. 75–151. MR 0485698
Martin Eichler, On theta functions of real algebraic number fields, Acta Arith. 33 (1977), no. 3, 269–292. MR 563061, DOI 10.4064/aa-33-3-269-292
Daniel File, Kimball Martin, and Ameya Pitale, Test vectors and central $L$-values for $\textrm {GL}(2)$, Algebra Number Theory 11 (2017), no. 2, 253–318. MR 3641876, DOI 10.2140/ant.2017.11.253
Ute Gebhardt, Explicit construction of spaces of Hilbert modular cusp forms using quaternionic theta series, Dissertation, 2009.
Stephen S. Gelbart, Automorphic forms on adèle groups, Annals of Mathematics Studies, No. 83, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. MR 0379375, DOI 10.1515/9781400881611
Benedict H. Gross, Local orders, root numbers, and modular curves, Amer. J. Math. 110 (1988), no. 6, 1153–1182. MR 970123, DOI 10.2307/2374689
H. Hijikata, A. Pizer, and T. Shemanske, Orders in quaternion algebras, J. Reine Angew. Math. 394 (1989), 59–106. MR 977435
Hiroaki Hijikata, Arnold K. Pizer, and Thomas R. Shemanske, The basis problem for modular forms on $\Gamma _0(N)$, Mem. Amer. Math. Soc. 82 (1989), no. 418, vi+159. MR 960090, DOI 10.1090/memo/0418
Hiroaki Hijikata and Hiroshi Saito, On the representability of modular forms by theta series, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 13–21. MR 0357332
H. Jacquet and R. P. Langlands, Automorphic forms on $\textrm {GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654, DOI 10.1007/BFb0058988
Kimball Martin, The Jacquet-Langlands correspondence, Eisenstein congruences, and integral $L$-values in weight 2, Math. Res. Lett. 24 (2017), no. 6, 1775–1795. MR 3762695, DOI 10.4310/MRL.2017.v24.n6.a11
Kimball Martin, Congruences for modular forms $\textrm {mod}\,2$ and quaternionic $S$-ideal classes, Canad. J. Math. 70 (2018), no. 5, 1076–1095. MR 3831915, DOI 10.4153/CJM-2017-019-1
B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287, DOI 10.1007/BF02684339
Arnold Pizer, An algorithm for computing modular forms on $\Gamma _{0}(N)$, J. Algebra 64 (1980), no. 2, 340–390. MR 579066, DOI 10.1016/0021-8693(80)90151-9
Arnold Pizer, Theta series and modular forms of level $p^{2}M$, Compositio Math. 40 (1980), no. 2, 177–241. MR 563541
Paul Ponomarev, Newforms of squarefree level and theta series, Math. Ann. 345 (2009), no. 1, 185–193. MR 2520057, DOI 10.1007/s00208-009-0348-2
Hiroshi Saito, On Tunnell’s formula for characters of $\textrm {GL}(2)$, Compositio Math. 85 (1993), no. 1, 99–108. MR 1199206
Thomas R. Shemanske and Lynne H. Walling, Twists of Hilbert modular forms, Trans. Amer. Math. Soc. 338 (1993), no. 1, 375–403. MR 1102225, DOI 10.1090/S0002-9947-1993-1102225-X
Hideo Shimizu, Theta series and automorphic forms on $\textrm {GL}_{2}$, J. Math. Soc. Japan 24 (1972), 638–683. MR 333081, DOI 10.2969/jmsj/02440638
Goro Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679. MR 507462
Goro Shimura, The arithmetic of certain zeta functions and automorphic forms on orthogonal groups, Ann. of Math. (2) 111 (1980), no. 2, 313–375. MR 569074, DOI 10.2307/1971202
Jerrold B. Tunnell, On the local Langlands conjecture for $GL(2)$, Invent. Math. 46 (1978), no. 2, 179–200. MR 476703, DOI 10.1007/BF01393255
Jerrold B. Tunnell, Local $\epsilon$-factors and characters of $\textrm {GL}(2)$, Amer. J. Math. 105 (1983), no. 6, 1277–1307. MR 721997, DOI 10.2307/2374441
J.-L. Waldspurger, Formes quadratiques à $4$ variables et relèvement, Acta Arith. 36 (1980), no. 4, 377–405 (French). MR 585893, DOI 10.4064/aa-36-4-377-405