AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Toric Periods and $p$-adic Families of Modular Forms of Half-Integral Weight
About this Title
V. Vatsal
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 289, Number 1438
ISBNs: 978-1-4704-6550-6 (print); 978-1-4704-7593-2 (online)
DOI: https://doi.org/10.1090/memo/1438
Published electronically: August 24, 2023
Keywords: automorphic forms,
metaplectic group,
toric periods,
test vectors
Table of Contents
Chapters
- Introduction
1. Part 1: Global periods: Fourier coefficients of half-integer weight forms
- 1. Preliminaries
- 2. Classical and automorphic forms of half-integer weight
- 3. Fourier coefficients of modular forms of half-integer weight
2. Part 2: Interpolation of the Fourier coefficients
- 4. Preliminaries and the basic formula
- 5. $\widetilde \Lambda$-adic Waldspurger packets, and non-vanishing of the $\widetilde \Lambda$-adic lift.
- 6. Specialization to trivial character, and the trivial zero
3. Part 3: Local periods: test vectors
- 7. Preliminaries and statement of results
- 8. $K$-types and the principal series
- 9. The depth zero supercuspidal representations of $GL_2({\mathbf {Q}}_p)$.
Abstract
The primary goal of this work is to construct $p$-adic families of modular forms of half-integral weight, by using Waldspurger’s automorphic framework to make the results as comprehensive and precise as possible. A secondary goal is to clarify the role of test vectors as defined by Gross-Prasad in the elucidation of general formulae for the Fourier coefficients of modular forms of half-integral weight in terms of toric periods of the corresponding modular forms of integral weight. As a consequence of our work, we develop a generalization of a classical formula due to Shintani, and make precise the conditions under which Shintani’s lift vanishes. We also give a number of results on test vectors for ramified representations which are of independent interest.- Avner Ash and Glenn Stevens, Modular forms in characteristic $l$ and special values of their $L$-functions, Duke Math. J. 53 (1986), no. 3, 849–868. MR 860675, DOI 10.1215/S0012-7094-86-05346-9
- Massimo Bertolini, Henri Darmon, and Kartik Prasanna, Generalized Heegner cycles and $p$-adic Rankin $L$-series, Duke Math. J. 162 (2013), no. 6, 1033–1148. With an appendix by Brian Conrad. MR 3053566, DOI 10.1215/00127094-2142056
- Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic $L$-functions and their derivatives, Ann. of Math. (2) 131 (1990), no. 1, 53–127. MR 1038358, DOI 10.2307/1971508
- 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
- Ehud Moshe Baruch and Zhengyu Mao, Central value of automorphic $L$-functions, Geom. Funct. Anal. 17 (2007), no. 2, 333–384. MR 2322488, DOI 10.1007/s00039-007-0601-3
- Daniel Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997. MR 1431508, DOI 10.1017/CBO9780511609572
- William Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314. MR 337789, DOI 10.1007/BF01428197
- William Casselman, The restriction of a representation of $\textrm {GL}_{2}(k)$ to $\textrm {GL}_{2}({\mathfrak {o}})$, Math. Ann. 206 (1973), 311–318. MR 338274, DOI 10.1007/BF01355984
- Li Cai, Jie Shu, and Ye Tian, Explicit Gross-Zagier and Waldspurger formulae, Algebra Number Theory 8 (2014), no. 10, 2523–2572. MR 3298547, DOI 10.2140/ant.2014.8.2523
- Christophe Cornut and Vinayak Vatsal, Nontriviality of Rankin-Selberg $L$-functions and CM points, $L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121–186. MR 2392354, DOI 10.1017/CBO9780511721267.005
- Henri Darmon and Gonzalo Tornaría, Stark-Heegner points and the Shimura correspondence, Compos. Math. 144 (2008), no. 5, 1155–1175. MR 2457522, DOI 10.1112/S0010437X08003552
- Yuval Z. Flicker, Automorphic forms on covering groups of $\textrm {GL}(2)$, Invent. Math. 57 (1980), no. 2, 119–182. MR 567194, DOI 10.1007/BF01390092
- 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
- Benedict H. Gross and Dipendra Prasad, Test vectors for linear forms, Math. Ann. 291 (1991), no. 2, 343–355. MR 1129372, DOI 10.1007/BF01445212
- S. Gelbart and I. Piatetski-Shapiro, Some remarks on metaplectic cusp forms and the correspondences of Shimura and Waldspurger, Israel J. Math. 44 (1983), no. 2, 97–126. MR 693355, DOI 10.1007/BF02760615
- Benedict H. Gross, Heights and the special values of $L$-series, Number theory (Montreal, Que., 1985) CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115–187. MR 894322
- 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
- Ralph Greenberg and Glenn Stevens, $p$-adic $L$-functions and $p$-adic periods of modular forms, Invent. Math. 111 (1993), no. 2, 407–447. MR 1198816, DOI 10.1007/BF01231294
- Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
- Florian Herzig, The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations, Duke Math. J. 149 (2009), no. 1, 37–116. MR 2541127, DOI 10.1215/00127094-2009-036
- Haruzo Hida, On $\Lambda$-adic forms of half integral weight for $\textrm {SL}(2)_{/\textbf {Q}}$, Number theory (Paris, 1992–1993) London Math. Soc. Lecture Note Ser., vol. 215, Cambridge Univ. Press, Cambridge, 1995, pp. 139–166. MR 1345178, DOI 10.1017/CBO9780511661990.010
- Yueke Hu, Jie Shu, and Hongbo Yin, Waldspurger’s period integral for newforms, Acta Arith. 195 (2020), no. 2, 177–197. MR 4109894, DOI 10.4064/aa190212-3-10
- Winfried Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32–72. MR 660784, DOI 10.1515/crll.1982.333.32
- Winfried Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237–268. MR 783554, DOI 10.1007/BF01455989
- Stephen S. Kudla, Michael Rapoport, and Tonghai Yang, Derivatives of Eisenstein series and Faltings heights, Compos. Math. 140 (2004), no. 4, 887–951. MR 2059224, DOI 10.1112/S0010437X03000459
- Zhengyu Mao, On a generalization of Gross’s formula, Math. Z. 271 (2012), no. 1-2, 593–609. MR 2917160, DOI 10.1007/s00209-011-0879-6
- Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. MR 1021004, DOI 10.1007/3-540-29593-3
- Kimball Martin and David Whitehouse, Central $L$-values and toric periods for $\textrm {GL}(2)$, Int. Math. Res. Not. IMRN 1 (2009), Art. ID rnn127, 141–191. MR 2471298, DOI 10.1093/imrn/rnn127
- Tadashi Ochiai, On the two-variable Iwasawa main conjecture, Compos. Math. 142 (2006), no. 5, 1157–1200. MR 2264660, DOI 10.1112/S0010437X06002223
- Olivier Fouquet and Tadashi Ochiai, Control theorems for Selmer groups of nearly ordinary deformations, J. Reine Angew. Math. 666 (2012), 163–187. MR 2920885, DOI 10.1515/CRELLE.2011.120
- Masami Ohta, On cohomology groups attached to towers of algebraic curves, J. Math. Soc. Japan 45 (1993), no. 1, 131–183. MR 1195688, DOI 10.2969/jmsj/04510131
- Masami Ohta, On the $p$-adic Eichler-Shimura isomorphism for $\Lambda$-adic cusp forms, J. Reine Angew. Math. 463 (1995), 49–98. MR 1332907, DOI 10.1515/crll.1995.463.49
- Shiv Prakash Patel and Dipendra Prasad, Multiplicity formula for restriction of representations of $\widetilde {\textrm {GL}_2}(F)$ to $\widetilde {\textrm {SL}_2}(F)$, Proc. Amer. Math. Soc. 144 (2016), no. 2, 903–908. MR 3430864, DOI 10.1090/proc12721
- D. Prasad, Notes on modular representations of $p$-adic groups and the Langlands correspondence, 2010.
- Ilya Piatetski-Shapiro, Complex representations of $\textrm {GL}(2,\,K)$ for finite fields $K$, Contemporary Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1983. MR 696772
- Nick Ramsey, $p$-Adic interpolation of square roots of central $L$-values of modular forms, Math. Ann. 358 (2014), no. 3-4, 1031–1058. MR 3175149, DOI 10.1007/s00208-013-0988-0
- Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
- Takuro Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126. MR 389772
- Allan J. Silberger, $\textrm {PGL}_{2}$ over the $p$-adics: its representations, spherical functions, and Fourier analysis, Lecture Notes in Mathematics, Vol. 166, Springer-Verlag, Berlin-New York, 1970. MR 285673
- Glenn Stevens, $\Lambda$-adic modular forms of half-integral weight and a $\Lambda$-adic Shintani lifting, Arithmetic geometry (Tempe, AZ, 1993) Contemp. Math., vol. 174, Amer. Math. Soc., Providence, RI, 1994, pp. 129–151. MR 1299739, DOI 10.1090/conm/174/01856
- C. M. Skinner and A. J. Wiles, Residually reducible representations and modular forms, Inst. Hautes Études Sci. Publ. Math. 89 (1999), 5–126 (2000). MR 1793414
- V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), no. 2, 397–419. MR 1695203, DOI 10.1215/S0012-7094-99-09811-3
- J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. (9) 59 (1980), no. 1, 1–132 (French). MR 577010
- J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (French). MR 646366
- J.-L. Waldspurger, Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173–242 (French). MR 783511
- Jean-Loup Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219–307 (French). MR 1103429, DOI 10.1515/form.1991.3.219