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Bounding Selmer Groups for the Rankin–Selberg Convolution of Coleman Families

Published online by Cambridge University Press:  17 July 2020

Andrew Graham*
Department of Mathematics, South Kensington Campus, Imperial College London, LondonSW7 2AZ, UK
Daniel R. Gulotta
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK e-mail:
Yujie Xu
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA02138 e-mail:


Let f and g be two cuspidal modular forms and let ${\mathcal {F}}$ be a Coleman family passing through f, defined over an open affinoid subdomain V of weight space $\mathcal {W}$ . Using ideas of Pottharst, under certain hypotheses on f and $g,$ we construct a coherent sheaf over $V \times \mathcal {W}$ that interpolates the Bloch–Kato Selmer group of the Rankin–Selberg convolution of two modular forms in the critical range (i.e, the range where the p-adic L-function $L_p$ interpolates critical values of the global L-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$ .

© Canadian Mathematical Society 2020

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The Winter School was supported by NSF grant DMS-1504537 and by the Clay Mathematics Institute. AG was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London and Imperial College London. DG was supported in part by Royal Society grant RP∖EA∖180020. YX was supported by Harvard University scholarships.


Andreatta, F. and Iovita, A., Triple product p-adic L-functions associated to finite slope p-adic families of modular forms, with an appendix by Eric Urban. arXiv:1708.02785 Google Scholar
Bellaïche, J., Critical $p$ -adic $L$ -functions. Invent. Math. 189(2012), no. 1, 160. CrossRefGoogle Scholar
Berger, L. and Colmez, P., Familles de représentations de de Rham et monodromie p-adique. Représentations p-adiques de groupes p-adiques I: Représentations galoisiennes et (φ,Γ)-modules, vol. 319, Astérisque, 2008, pp. 303337.Google Scholar
Berger, L., Représentations $p$ -adiques et équations différentielles. Invent. Math. 148(2002), no. 2, 219284. CrossRefGoogle Scholar
Berger, L., Bloch and Kato’s exponential map: three explicit formulas. Doc. Math (2003), Extra Vol., 99129. Google Scholar
Coleman, R. F., Classical and overconvergent modular forms. Invent. Math 124(1996), no. 1-3, 215241.CrossRefGoogle Scholar
Colmez, P., Représentations triangulines de dimension 2. Astérisque (2008), no. 319, 213258.Google Scholar
Colmez, P., Fonctions d’une variable $p$ -adique. Astérisque 330(2010), 1359. MR 2642404Google Scholar
Jannsen, U., Continuous étale cohomology. Math. Ann. 280(1988), no. 2, 207245. CrossRefGoogle Scholar
Kedlaya, K. S. and Liu, R., On families of $(\varphi, \varGamma )$ -modules. Algebra. Number Theory 4(2010), no. 7, 943967. CrossRefGoogle Scholar
Kedlaya, K. S., Pottharst, J., and Xiao, L., Cohomology of arithmetic families of $(\varphi, \varGamma )$ -modules. J. Amer. Math. Soc. 27(2014), no. 4, 10431115. CrossRefGoogle Scholar
Kings, G., On $p$ -adic interpolation of motivic Eisenstein classes. Elliptic curves, modular forms and Iwasawa theory, Springer Proc. Math. Stat, 188, Springer, Cham, 2016, pp. 335371.Google Scholar
Kings, G., Loeffler, D., and Zerbes, S. Livia, Rankin–Eisenstein classes for modular forms. Amer. J. Math (2015), to appear. Google Scholar
Kings, G., Loeffler, D., and Zerbes, S. Livia, Rankin-Eisenstein classes and explicit reciprocity laws. Camb. J. Math. 5(2017), no. 1, 1122. 3230818CrossRefGoogle Scholar
Lang, S., Introduction to modular forms. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 222, Springer-Verlag, Berlin, 1995, with appendixes by D. Zagier and Walter Feit. Corrected reprint of the 1976 original.Google Scholar
Lei, A., Loeffler, D., and Zerbes, S., Coleman maps and the p-adic regulator. Algebra. Number Theory 5(2011), no. 8, 10951131. CrossRefGoogle Scholar
Lei, A., Loeffler, D., and Zerbes, S. L., Euler systems for Rankin-Selberg convolutions of modular forms. Ann. of Math. (2) 180(2014), no. 2, 653771. CrossRefGoogle Scholar
Liu, R., Cohomology and duality for $(\phi, \varGamma )$ -modules over the Robba ring. Int. Math. Res. Not. IMRN (2008), no. 3, Art. ID rnm150. Google Scholar
Liu, R., Triangulation of refined families. Comment. Math. Helv. 90(2015), no. 4, 831904. CrossRefGoogle Scholar
Loeffler, D., Images of adelic Galois representations for modular forms. Glasg. Math. J. 59(2017), no. 1, 1125. CrossRefGoogle Scholar
Loeffler, D., A Note on p-adic Rankin–Selberg L-functions. Canad. Math. Bull 61 (2018), no. 3, 608621. CrossRefGoogle Scholar
Loeffler, D. and Weinstein, J., On the computation of local components of a newform. Math. Comp. 81(2012), no. 278, 11791200. CrossRefGoogle Scholar
Loeffler, D. and Zerbes, S. L., Rankin-Eisenstein classes in Coleman families. Res. Math. Sci. 3(2016), Art. ID 29. CrossRefGoogle Scholar
Nakamura, K., Iwasawa theory of de Rham $(\varphi, \varGamma )$ -modules over the Robba ring. J. Inst. Math. Jussieu 13(2014), no. 1, 65118. CrossRefGoogle Scholar
Nekovář, J., Selmer complexes. Astérisque (2006), no. 310, viii+559.Google Scholar
Perrin-Riou, B., $p$ -adic $L$ -functions and $p$ -adic representations. SMF/AMS Texts and Monographs, vol. 3, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2000, Translated from the 1995 French original by Leila Schneps and revised by the author.Google Scholar
Pottharst, J., Cyclotomic Iwasawa theory of motives. Preprint, 2012.Google Scholar
Pottharst, J., Analytic families of finite-slope Selmer groups. Algebra. Number Theory 7(2013), no. 7, 15711612. CrossRefGoogle Scholar
Rubin, K., Euler systems. Ann. of Math., 147, Princeton University Press, Princeton, NJ, 2000, Google Scholar
Urban, E., Nearly overconvergent modular forms. Iwasawa Theory 2012 (Berlin, Heidelberg), Springer Berlin Heidelberg, 2014, pp. 401441.Google Scholar