Bounds for coefficients of cusp forms and extremal lattices

Jenkins, Paul A.; Rouse, Jeremy

doi:10.1112/blms/bdr030

Cited by 22 publications

(28 citation statements)

References 11 publications

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“…One class of examples is given by free bosons compactified on extremal lattices. Such lattices can be explicitly constructed for small central charge and are known not to exist for c ≥ 163264 [33]. Appealing to more standard string theory examples, we also consider a gravitational theory in flat space, and discuss the D1-D5 system in highly curved AdS space.…”

Section: Large Gap Examplesmentioning

confidence: 99%

Universal bounds on charged states in 2d CFT and 3d gravity

Benjamin¹,

Dyer²,

Fitzpatrick

et al. 2016

J. High Energ. Phys.

108

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Abstract:We derive an explicit bound on the dimension of the lightest charged state in two dimensional conformal field theories with a global abelian symmetry. We find that the bound scales with c and provide examples that parametrically saturate this bound. We also prove that any such theory must contain a state with charge-to-mass ratio above a minimal lower bound. We comment on the implications for charged states in three dimensional theories of gravity.

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Section: Large Gap Examplesmentioning

confidence: 99%

Universal bounds on charged states in 2d CFT and 3d gravity

Benjamin¹,

Dyer²,

Fitzpatrick

et al. 2016

J. High Energ. Phys.

108

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“…We now state a useful recent result of Jenkins and Rouse [5] which gives an estimate for the Fourier coefficients of a cusp form f = n 1 a f (n)q n of level one in terms of its first d k Fourier coefficients:…”

Section: Basic Definitions and Resultsmentioning

confidence: 99%

Linear relations among Poincaré series

Das¹,

Ganguly

2012

Bulletin of the London Mathematical Society

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We study general linear relations among Poincaré series of different indices with arbitrary but fixed level and weight. We establish several conditions the coefficients involved must satisfy. In particular, we show under some mild assumptions that two Poincaré series of distinct indices cannot be identically equal unless they are both identically zero.(i) for which n, if any, is the Poincaré series P n identically zero? and, (ii) for which pairs (m, n) of distinct positive integers, if any, do we have P m ≡ P n ?We shall see that these two questions are intimately related. Choie, Kohnen, and Ono [2] and Rhoades [13] have previously studied linear relations among Poincaré series, relating the coefficients c i to the principal parts of certain weakly holomorphic modular forms for SL(2, Z). Here we are concerned with determining the algebraic and arithmetic nature of the coefficients rather than giving an explicit description of them.Two of our principal results are the following.Theorem 1.1. Suppose we have P n,k,q ≡ N i=1 c i P ni,k,q , where P n,k,q is not identically zero and (n, q) = 1. If each c i is an algebraic integer, then n (d(n)) 2/(k−1) (n, n 1 n 2 . . . n N ) 2 , where d(n) is the number of divisors of n.

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“…For the remainder of the section, we set v = 1.16 and y = .865, so that τ = u+1.16i, z = x+.865i, where u, x ∈ [−1/2, 1/2]. This choice of v, y is identical to that in [17]. These values of v and y give reasonable bounds, and keep the difference of Hauptmoduln in the denominator of (5.1) far enough from zero.…”

Section: Standard Computations Then Show Thatmentioning

confidence: 99%

“…[19], [27], [28]). Rouse and the first author [17] gave an explicit bound on the implied constant for all cusp forms for SL 2 (Z) (earlier work of Chua [5] makes Hecke's O(n k 2 bound explicit). For a cusp form 1 G = ∞ n=1 a(n)q n of weight k for SL 2 (Z), they proved that |a(n)| ≤ log k where ℓ is the dimension of the weight k cusp form space S k (SL 2 (Z)).…”

Section: Introductionmentioning

confidence: 99%