The asymptotic distribution of traces of Maass–Poincaré series

Folsom, Amanda; Masri, Riad

doi:10.1016/j.aim.2010.11.001

Cited by 6 publications

(4 citation statements)

References 28 publications

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“…Part of this analysis requires an explicit upper bound for the L 2 -norm of iterates of the hyperbolic Laplacian applied to a regularized version of F ε (z). For more details concerning these arguments, we refer the reader to [7]. Finally, we remark that the problem of estimating p(n) can be thought of as a special case of the problem of estimating the coefficients of negative half-integral weight weakly holomorphic modular forms and harmonic Maass forms.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

“…[20, section 11]), the strategy in this paper should apply to the problem of bounding the analog of Rademacher's error term in these situations (see e.g. [7]). …”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

“…for some sufficiently large integer a 1 > 0 (see [7]). Then by combining the estimates (5.4) and (5.5), we obtain 1<Im(z Q )≤1+ε…”

Section: Proposition 42 For Ymentioning

confidence: 99%

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Equidistribution of Heegner points and the partition function

Folsom

Masri

2010

Math. Ann.

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Let p(n) denote the number of partitions of a positive integer n. In this paper we study the asymptotic growth of p(n) using the equidistribution of Galois orbits of Heegner points on the modular curve X 0 (6). We obtain a new asymptotic formula for p(n) with an effective error term which is O(n −( 1 2 +δ) ) for some δ > 0. We then use this asymptotic formula to sharpen the classical bounds of Hardy and Ramanujan, Rademacher, and Lehmer on the error term in Rademacher's exact formula for p(n).

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Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

“…[20, section 11]), the strategy in this paper should apply to the problem of bounding the analog of Rademacher's error term in these situations (see e.g. [7]). …”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

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Equidistribution of Heegner points and the partition function

Folsom

Masri

2010

Math. Ann.

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“…One example is the function F d,k appearing in the average formula (2.2). Other examples occur in the work of Duke [6] on traces of singular moduli, and the work of Folsom and the second author on the limiting distribution of traces of Maass-Poincaré series [8], and the asymptotic distribution of the partition function [9].…”

Section: Galois Suborbits Of Heegner Pointsmentioning

confidence: 99%

Nonvanishing of Hecke L-functions and the Bloch–Kato conjecture

2010

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In this paper we study the central values of L-functions associated to a large class of algebraic Hecke characters of imaginary quadratic fields. When these central values are nonzero, the Bloch-Kato conjecture predicts an exact formula for the algebraic parts of the central values in terms of periods and arithmetic data, most notably the Selmer groups corresponding to the Hecke characters. We investigate the nonvanishing of these central values, and prove the p-part of the Bloch-Kato conjecture in these cases for primes p which split in K .

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Nonvanishing of Hecke L-Functions for CM Fields and Ranks of Abelian Varieties

Masri

Yang

2011

Geom. Funct. Anal.

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In this paper we prove a nonvanishing theorem for central values of L-functions associated to a large class of algebraic Hecke characters of CM number fields. A key ingredient in the proof is an asymptotic formula for the average of these central values. We combine the nonvanishing theorem with work of Tian and Zhang [TiZ] to deduce that infinitely many of the CM abelian varieties associated to these Hecke characters have Mordell-Weil rank zero. Included among these abelian varieties are higher-dimensional analogues of the elliptic Q-curves A(D) of B. Gross [Gr].

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The asymptotic distribution of traces of Maass–Poincaré series

Cited by 6 publications

References 28 publications

Equidistribution of Heegner points and the partition function

Equidistribution of Heegner points and the partition function

Nonvanishing of Hecke L-functions and the Bloch–Kato conjecture

Nonvanishing of Hecke L-Functions for CM Fields and Ranks of Abelian Varieties

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