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THETA BLOCK FOURIER EXPANSIONS, BORCHERDS PRODUCTS AND A SEQUENCE OF NEWMAN AND SHANKS

  • CRIS POOR (a1), JERRY SHURMAN (a2) and DAVID S. YUEN (a3)

Abstract

The ‘Borcherds products everywhere’ construction [Gritsenko et al., ‘Borcherds products everywhere’, J. Number Theory 148 (2015), 164–195] creates paramodular Borcherds products from certain theta blocks. We prove that the $q$ -order of every such Borcherds product lies in a sequence  $\{C_{\unicode[STIX]{x1D708}}\}$ , depending only on the $q$ -order  $\unicode[STIX]{x1D708}$ of the theta block. Similarly, the $q$ -order of the leading Fourier–Jacobi coefficient of every such Borcherds product lies in a sequence  $\{A_{\unicode[STIX]{x1D708}}\}$ , and this is the sequence  $\{a_{n}\}$ from work of Newman and Shanks in connection with a family of series for  $\unicode[STIX]{x1D70B}$ . Our proofs use a combinatorial formula giving the Fourier expansion of any theta block in terms of its germ.

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[1] Bruinier, J. H., Borcherds Products on O (2, l) and Chern Classes of Heegner Divisors, Lecture Notes in Mathematics, 1780 (Springer, Berlin, 2002).
[2] Eichler, M. and Zagier, D., The Theory of Jacobi Forms, Progress in Mathematics, 55 (Birkhäuser Boston, Boston, MA, 1985).
[3] Gritsenko, V. A. and Nikulin, V. V., ‘Automorphic forms and Lorentzian Kac–Moody algebras. II’, Internat. J. Math. 9(2) (1998), 201275.
[4] Gritsenko, V. A., Poor, C. and Yuen, D. S., ‘Borcherds products everywhere’, J. Number Theory 148 (2015), 164195.
[5] Gritsenko, V. A., Skoruppa, N.-P. and Zagier, D., ‘Theta blocks’, in preparation.
[6] Newman, M. and Shanks, D., ‘On a sequence arising in series for 𝜋’, Math. Comp. 42(1655) (1984), 397463.
[7] Poor, C., Shurman, J. and Yuen, D. S., ‘Finding all Borcherds lift paramodular cusp forms of a given weight and level’, preprint, arXiv:1803.11092.
[8] Shanks, D., ‘Dihedral quartic approximations and series for 𝜋’, J. Number Theory 14 (1982), 397423.
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THETA BLOCK FOURIER EXPANSIONS, BORCHERDS PRODUCTS AND A SEQUENCE OF NEWMAN AND SHANKS

  • CRIS POOR (a1), JERRY SHURMAN (a2) and DAVID S. YUEN (a3)

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