Soumya Das

Associate Professor, Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India.

email: firstname.math.20@gmail.com,

firstname@iisc.ac.in

~If someone does science in a forest, but doesn't make a sound, did science actually happen?

- Michael Short.

Research interests: Analytic aspects of Automorphic forms, analytic and algebraic number theory.

Teaching

Aug 2023: MA 317

Introductory course in analytic number theory. We have covered so far: Arithmetic functions and their orders, various summation formulas: Euler-Maclaurin, Poisson etc., general theory of Dirichlet series, Dirichlet L-functions, primitive characters, Analytic continuation of Riemann zeta and Dirichlet L-functions, proof of Dirichlet's theorem on primes in A.P., Quadratic characters-fundamental discriminants and Gauss sums, non-vanishing of L-fns on Re(s)=1 and prime number theorem (weak form), Infinite products, Weirstrass-Hadamard factorisation of entire functions, Applications to Riemann zeta and Dirichlet L-fns, Ikehara-Wiener theorem, Stirling's formula and analytic properties of the Gamma function in detail, zeros of L-fns: classical zero-free regions -- Landau-Siegel zero, Mellin transform and inversion, Perron's formula, explicit formulae, bounds on L-fns, PNT with error term, various relations among deep conjectures, class number formula for Dedekind zeta.

Appendix: Fourier theory

Jan 2023: MA 350

This course is a sequel to MA 317 (click HERE to send a request to join), and only those who have either attended it or are comfortable with its contents are encouraged to attend. The contents actually covered in MA 317 can be found from my website. We will cover Bombieri-Vinogradov theorem and PNT for A.P.s, Large sieve, Circle method (with emphasis on Diophantine equations, quadratic forms), exponential sums and integrals, zero-density estimates, applications of sieve methods, various results pertaining to Landau-Siegel zeros, Sub-convexity estimates in various situations, detecting zeros (infinitely many zeros on 1/2 line) and other things if time permits.

Jan 2022: MA 313, Algebraic number theory, pre-requisites: MA 213 or equivalent.

See Teams page for more details, or write an email.

Jan 2021: MA 215, Introduction to Modular forms:

These would be expanded and polished extensively -- but you are strongly advised to consult these notes to judge it's content before deciding to opt for the course.

The meeting link can be found from the Maths dept. Intranet.

Send an email to me if you want to attend.

Meeting: Tuesday-Thursday, 3.30-5.00 pm.

**Practice problems should be considered as supplements to the class, and are examinable, unless mentioned otherwise. You can ask about any of them to either the T.A. or the instructor.

***Final exam would be held on 11 June , from 2 PM to 5 PM. Venue would be .... Please write short answers. Short and correct answers would be given high value. In particular there is no point in deriving or proving things which have already been proved in class. You may just state them.

(a) Practice problems: 10%

(b) Quizzes: 30%

(c) Mid-term: 30%

(d) Final-term: 30%

## Preprints

Fourier coefficients and cuspidality of modular forms: a new approach, https://arxiv.org/abs/2407.15222

L^\infty-sizes of the spaces Siegel cusp forms of degree n via Poincaré series. https://arxiv.org/abs/2406.19335

New and old Saito-Kurokawa lifts classically via L^2 norms and bounds on their supnorms: level aspect. https://arxiv.org/pdf/2403.17401.pdf (with P. Anamby)

Bounds for the Bergman kernel and the sup-norm of holomorphic Siegel cusp forms. (with H. Krishna). IMRN, to appear. (provided for the first time the expected size of degree 2 Siegel cuspforms.)

On the sup-norm of Maass lifts. RMS lecture Notes (2020) for conference proceedings at IIT ROPAR, (with J. Sengupta).

## Publications

Jacobi forms, Saito-Kurokawa lifts, their Pullbacks and sup-norms on average. (with P. Anamby). https://arxiv.org/abs/2206.02192, 52 pp. Res. Math. Sci.

On fundamental Fourier coefficients of Siegel modular forms. (with S. Böcherer), 48 pp. J. Inst. Math. Jussieu, https://doi.org/10.1017/S1474748021000086

arXiv, 52 pp.: https://arxiv.org/abs/1810.00762 has an Appendix in addition.

(these versions prove Andrianov's conjecture on the functional equation of the spinor L-function for holomorphic Siegel cusp forms of degree 3 with full level, when combined with the work of A. Pollack.)On Fourier coefficients of elliptic modular forms mod l with applications to Siegel modular forms (with. S. Böcherer), 32pp. Manuscripta Math., https://doi.org/10.1007/s00229-021-01277-8

Large Hecke eigenvalues and an Omega result for non Saito--Kurokawa lifts (with P. Anamby, R. Pal), 13pp. Ramanujan J., to appear.

Omega results for Fourier coefficients of half-integral weight and Siegel modular forms, 14pp. to appear in Springer Conf. Proceedings (intended for the conference proc. at KSOM commemorating the 60th birthday of Prof. M. Manickam).

Hecke-Siegel type threshold for square-free Fourier coefficients: an improvement RIMS Kokyuroku Kyoto Univ., proceedings for the conference “Analytic and Arithmetic Theory of Automorphic Forms”, January 2018. (With P. Anamby)

MR4034598 Das, Soumya; Jha, Abhash Kumar Analytic properties of twisted real-analytic Hermitian Klingen type Eisenstein series and applications. Abh. Math. Semin. Univ. Hambg. 89 (2019), no. 2, 105–116. 11F55 (11F66)

MR3974676 Das, Soumya; Pal, Ritwik The first negative eigenvalue of Yoshida lifts. Res. Number Theory 5 (2019), no. 3, Paper No. 20, 9 pp. (Reviewer: Hideshi Takayanagi) 11F46 (11F66)

MR3908796 Anamby, Pramath; Das, Soumya Distinguishing Hermitian cusp forms of degree 2 by a certain subset of all Fourier coefficients. Publ. Mat. 63 (2019), no. 1, 307–341. (Reviewer: Brundaban Sahu) 11F30 (11F50 11F55)

MR3859224 Das, Soumya; Khan, Rizwanur The third moment of symmetric square L-functions. Q. J. Math. 69 (2018), no. 3, 1063–1087. (Reviewer: Valentin Blomer) 11M06

MR3846155 Böcherer, Siegfried; Das, Soumya Petersson norms of not necessarily cuspidal Jacobi modular forms and applications. Adv. Math. 336 (2018), 335–376. (Reviewer: G. K. Sankaran) 11F50 (11F46)

MR3836146 Böcherer, Siegfried; Das, Soumya Cuspidality and the growth of Fourier coefficients of modular forms. J. Reine Angew. Math. 741 (2018), 161–178. (Reviewer: Hideshi Takayanagi) 11F30 (11F46)

MR3825472 Anamby, Pramath; Das, Soumya Bounds for the Petersson norms of the pullbacks of Saito-Kurokawa lifts. J. Number Theory 191 (2018), 289–304. (Reviewer: Karam Deo Shankhadhar) 11F11 (11F46 11F66)

MR3798786 Das, Soumya; Kohnen, Winfried On sign changes of eigenvalues of Siegel cusp forms of genus 2 in prime powers. Acta Arith. 183 (2018), no. 2, 167–172. (Reviewer: Tomoyoshi Ibukiyama) 11F46 (11F41)

MR3931445 Anamby, Pramath; Das, Soumya Sturm-like bound for square-free Fourier coefficients. L-functions and automorphic forms, 1–7, Contrib. Math. Comput. Sci., 10, Springer, Cham, 2017. 11F30

MR3657159 Das, Soumya; Pal, Ritwik Jacobi forms and differential operators: odd weights. J. Number Theory 179 (2017), 113–125. (Reviewer: Karam Deo Shankhadhar) 11F50 (11F11 11F25)

MR3489079 Böcherer, Siegfried; Das, Soumya Cuspidality and the growth of Fourier coefficients: small weights. Math. Z. 283 (2016), no. 1-2, 539–553. (Reviewer: Martin Raum) 11F30 (11F46)

MR3477807 Das, Soumya Cuspidality and the growth of Fourier coefficients of modular forms: a survey. Indian J. Pure Appl. Math. 47 (2016), no. 1, 9–22. (Reviewer: Kalyan Chakraborty) 11F30

MR3477047 Das, Soumya; Ganguly, Satadal A note on small gaps between nonzero Fourier coefficients of cusp forms. Proc. Amer. Math. Soc. 144 (2016), no. 6, 2301–2305. (Reviewer: Cherng-tiao Perng) 11F30 (11F11 11G05)

MR3385236 Das, Soumya; Khan, Rizwanur Simultaneous nonvanishing of Dirichlet L-functions and twists of Hecke-Maass L-functions. J. Ramanujan Math. Soc. 30 (2015), no. 3, 237–250. (Reviewer: Sanoli Gun) 11F11 (11F66)

MR3366849 Das, Soumya; Kohnen, Winfried Nonvanishing of Koecher-Maass series attached to Siegel cusp forms. Adv. Math. 281 (2015), 624–669. (Reviewer: Valentin Blomer) 11F30 (11F46)

MR3365786 Das, Soumya; Sengupta, Jyoti L∞ norms of holomorphic modular forms in the case of compact quotient. Forum Math. 27 (2015), no. 4, 1987–2001. (Reviewer: Guangshi Lü) 11F11 (11F12) (update: use this version PDF)

MR3327845 Das, Soumya; Meher, Jaban On quasimodular eigenforms. Int. J. Number Theory 11 (2015), no. 3, 835–842. (Reviewer: Dominic A. Lanphier) 11F30 (11F11)

MR3296015 Das, Soumya; Ramakrishnan, B. Jacobi forms and differential operators. J. Number Theory 149 (2015), 351–367. (Reviewer: John Francis Robert Duncan) 11F50 (11F11 11F25)

MR3280046 Böcherer, Siegfried; Das, Soumya Linear independence of Poincaré series of exponential type via non-analytic methods. Trans. Amer. Math. Soc. 367 (2015), no. 2, 1329–1345. (Reviewer: Sheng-Chi Liu) 11F30 (11F46)

MR3251716 Das, Soumya; Ganguly, Satadal Gaps between nonzero Fourier coefficients of cusp forms. Proc. Amer. Math. Soc. 142 (2014), no. 11, 3747–3755. (Reviewer: Scott Ahlgren) 11F30 (11F11 11N25)

MR3201897 Böcherer, Siegfried; Das, Soumya Characterization of Siegel cusp forms by the growth of their Fourier coefficients. Math. Ann. 359 (2014), no. 1-2, 169–188. (Reviewer: Rainer Schulze-Pillot) 11F30 (11F46)

MR3182539 Das, Soumya; Kohnen, Winfried; Sengupta, Jyoti On a convolution series attached to a Siegel Hecke cusp form of degree 2. Ramanujan J. 33 (2014), no. 3, 367–378. (Reviewer: Suzanne Caulk) 11F46 (11F66)

MR3148511 Das, Soumya; Sengupta, Jyoti An Omega-result for Saito-Kurokawa lifts. Proc. Amer. Math. Soc. 142 (2014), no. 3, 761–764. (Reviewer: B. Ramakrishnan) 11F46 (11F30)

MR3221309 Das, Soumya; Kohnen, Winfried Some remarks on the Resnikoff-Saldaña conjecture. The legacy of Srinivasa Ramanujan, 153–161, Ramanujan Math. Soc. Lect. Notes Ser., 20, Ramanujan Math. Soc., Mysore, 2013. (Reviewer: Jae-Hyun Yang) 11F30 (11F46)

MR3065087 Böcherer, Siegfried; Das, Soumya On holomorphic differential operators equivariant for the inclusion of Sp(n,R) in U(n,n). Int. Math. Res. Not. IMRN 2013, no. 11, 2534–2567. (Reviewer: Rolf Berndt) 32M15 (11F03 32W50)

MR2997487 Das, Soumya On the natural densities of eigenvalues of a Siegel cusp form of degree 2. Int. J. Number Theory 9 (2013), no. 1, 9–15. (Reviewer: Suzanne Caulk) 11F46 (11F30)

MR2997486 Das, Soumya; Ganguly, Satadal Nonvanishing of Poincaré series on average. Int. J. Number Theory 9 (2013), no. 1, 1–8. (Reviewer: Barış Kendirli) 11F11 (11F30)

MR2997572 Das, Soumya; Kohnen, Winfried; Sengupta, Jyoti Nonvanishing of Siegel-Poincaré series II. Acta Arith. 156 (2012), no. 1, 75–81. (Reviewer: Ellen E. Eischen) 11F46 (11F30)

MR2995143 Das, Soumya; Sengupta, Jyoti Nonvanishing of Siegel Poincaré series. Math. Z. 272 (2012), no. 3-4, 869–883. (Reviewer: Ellen E. Eischen) 11F46 (11L05)

MR2975157 Das, Soumya; Ganguly, Satadal Linear relations among Poincaré series. Bull. Lond. Math. Soc. 44 (2012), no. 5, 988–1000. (Reviewer: SoYoung Choi) 11F11 (11F30)

MR2769134 Das, Soumya Nonvanishing of Jacobi Poincaré series. J. Aust. Math. Soc. 89 (2010), no. 2, 165–179. (Reviewer: Olav K. Richter) 11F50 (11F30)

MR2738862 Das, Soumya Some aspects of Hermitian Jacobi forms. Arch. Math. (Basel) 95 (2010), no. 5, 423–437. (Reviewer: Anton Deitmar) 11F50 (11F60)

MR2723724 Das, Soumya Note on Hermitian Jacobi forms. Tsukuba J. Math. 34 (2010), no. 1, 59–78. (Reviewer: B. Ramakrishnan) 11F50 (11F55)