A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus

Banerjee, Debargha; Mandal, Tathagata

doi:10.1090/proc/14887

A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus
HTML articles powered by AMS MathViewer

by Debargha Banerjee and Tathagata Mandal PDF

Proc. Amer. Math. Soc. 148 (2020), 1509-1525 Request permission

Abstract:

In this article, we investigate the variance of the local $\varepsilon$-factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti [Proc. Amer. Math. Soc. 141 (2013), pp. 2615–2628]. For ramified principal series (with $p \| N$ and $p$ odd) and unramified supercuspidal representations of level zero, we relate these numbers with Morita’s $p$-adic Gamma function.

References

A. O. L. Atkin and Wen Ch’ing Winnie Li, Twists of newforms and pseudo-eigenvalues of $W$-operators, Invent. Math. 48 (1978), no. 3, 221–243. MR 508986, DOI 10.1007/BF01390245
Debargha Banerjee and Tathagata Mandal, Supercuspidal ramifications and traces of adjoint lifts, J. Number Theory 201 (2019), 292–321. MR 3958053, DOI 10.1016/j.jnt.2019.02.017
Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams, Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR 1625181
Shalini Bhattacharya and Eknath Ghate, Supercuspidal ramification of modular endomorphism algebras, Proc. Amer. Math. Soc. 143 (2015), no. 11, 4669–4684. MR 3391026, DOI 10.1090/proc/12629
Sazzad Ali Biswas, Langlands lambda function for quadratic tamely ramified extensions, J. Algebra Appl. 18 (2019), no. 7, 1950132, 10. MR 3977793, DOI 10.1142/S0219498819501329
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
J. W. S. Cassels and A. Fröhlich, Algebraic number theory, 2nd Edition, London Mathematical Society, 2010.
P. Deligne, Formes modulaires et représentations de $\textrm {GL}(2)$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 55–105 (French). MR 0347738
P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 501–597 (French). MR 0349635
L. Dieulefait, A. Pacetti, and P. Tsaknias, On the number of Galois orbits of newforms, https://arxiv.org/pdf/1805.10361.pdf.
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
Daniel Kohen and Ariel Pacetti, On Heegner points for primes of additive reduction ramifying in the base field, Trans. Amer. Math. Soc. 370 (2018), no. 2, 911–926. With an appendix by Marc Masdeu. MR 3729491, DOI 10.1090/tran/6990
R. P. Langlands, On the functional equation of the Artin l-functions, unpublished article, https://publications.ias.edu/sites/default/files/a-ps.pdf.
Rudolf Lidl and Harald Niederreiter, Introduction to finite fields and their applications, 1st ed., Cambridge University Press, Cambridge, 1994. MR 1294139, DOI 10.1017/CBO9781139172769
David Loeffler and Jared Weinstein, On the computation of local components of a newform, Math. Comp. 81 (2012), no. 278, 1179–1200. MR 2869056, DOI 10.1090/S0025-5718-2011-02530-5
Ariel Pacetti, On the change of root numbers under twisting and applications, Proc. Amer. Math. Soc. 141 (2013), no. 8, 2615–2628. MR 3056552, DOI 10.1090/S0002-9939-2013-11532-7
Anna Rio, Dyadic exercises for octahedral extensions. II, J. Number Theory 118 (2006), no. 2, 172–188. MR 2223979, DOI 10.1016/j.jnt.2005.08.007
Alain M. Robert, A course in $p$-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000. MR 1760253, DOI 10.1007/978-1-4757-3254-2
Ralf Schmidt, Some remarks on local newforms for $\rm GL(2)$, J. Ramanujan Math. Soc. 17 (2002), no. 2, 115–147. MR 1913897
J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607
J. T. Tate, Local constants, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 89–131. Prepared in collaboration with C. J. Bushnell and M. J. Taylor. MR 0457408
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
André Weil, Exercices dyadiques, Invent. Math. 27 (1974), 1–22 (French). MR 379445, DOI 10.1007/BF01389962
Kit Ming Yeung, On congruences for binomial coefficients, J. Number Theory 33 (1989), no. 1, 1–17. MR 1014384, DOI 10.1016/0022-314X(89)90056-5