Eulerian series as modular forms

By Kathrin Bringmann, Ken Ichi Ono, Robert C. Rhoades. 2007

92 citations recorded by the source corpus.

In 1988, Hickerson proved the celebrated “mock theta conjectures”, a collection of ten identities from Ramanujan’s “lost notebook” which express certain modular forms as linear combinations of mock theta functions. In the context of Maass forms, these identities arise from the peculiar phenomenon that two different harmonic Maass forms may have the same non-holomorphic parts. Using this perspective, we construct several infinite families of modular forms which are differences of mock theta functions.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8f574b61315ccd8f34732ad5d18eccf2b8bc4662

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

The basic bilateral hypergeometric series and the mock theta functions (2011, methodology): “Rhoades [ 6, Theorem 1.3 ] who used the theory of Harmonic weak Maass forms.”
RANK DIFFERENCES FOR OVERPARTITIONS (2007, background): “5] can be shown in many cases (such as when −d is not a square modulo ℓ) to be weakly holomorphic modular forms. This has been carried out in detail for the rank diﬀerences for ordinary partitions in [7]. For now, however, it seems that the groups involved are too small (and the number of inequivale”
Unearthing the visions of a master: harmonic Maass forms and number theory (2008, background): “In Section 10 we discuss work by the author, Bringmann, and R. Rhoades [57] on Eulerian series as modular forms.”
Automorphic Properties of Generating Functions for Generalized Rank Moments and Durfee Symbols (2008, background): “5 (for z 6= 1) could be concluded from [17] but for the readers convenience we give a proof here.”
Congruences for Andrews' Smallest Parts Partition Function and New Congruences for Dyson's Rank (2007, background): “is a weakly holomorphic weight 1=2 modular form when 1 24d is a quadratic nonresidue mod ‘ and when it is a quadratic residue under certain additional conditions on r1 and r2. As noted in [ 16 ] this was observed earlier by Atkin and Swinnerton-Dyer [11] in the cases ‘ = 5 and ‘ = 7. It should also ”
DYSON'S RANK, OVERPARTITIONS, AND WEAK MAASS FORMS (2007, methodology): “In view of Theorem 1.1 one can obtain results on overpartitions by arguing as in work of Ono and the first author [5, 6, 8, 9, 10 ].”
Heegner divisors, $L$-functions and harmonic weak Maass forms (2007, background): “Such forms have been a source of recent interest due to their connection to Ramanujan’s mock theta functions (see [BO1, BO2, BOR , O2, Za3, Zw1, Zw2]).”
Mock Jacobi forms in basic hypergeometric series (2008, background): “Proposition 5.2. [ 8, Theorem 1.3 ] Let K ′ and K ′′ be defined as in Theorem 1.5.”
A framework of Rogers–Ramanujan identities and their arithmetic properties (2014, background): “linked to algebraic geometry [23, 37], K-theory [28], conformal field theory [10, 44, 52], group theory [32], Kac–Moody, Virasoro, vertex and double affine Hecke algebras [25, 31, 50, 51, 53, 54, 55, 56], knot theory [6, 39, 40], modular forms [16, 17, 18, 19, 20, 21, 22, 63], orthogonal polynomials”
MOCK THETA FUNCTIONS AND QUANTUM MODULAR FORMS (2013, methodology): “3 after first proving a lemma about the convergence of relevant q-hypergeometric series at roots of unity, and after recalling important results of Lawrence and Zagier [31] and Bringmann and the second two authors [17].”