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This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.

Real zeros of holomorphic Hecke cusp forms and sieving short intervals

Kaisa Matomäki (2016)

Journal of the European Mathematical Society

We study so-called real zeros of holomorphic Hecke cusp forms, that is, zeros on three geodesic segments on which the cusp form (or a multiple of it) takes real values. Ghosh and Sarnak, who were the first to study this problem, showed the existence of many such zeros if many short intervals contain numbers whose prime factors all belong to a certain subset of the primes.We prove new results concerning this sieving problem which leads to improved lower bounds for the number of real zeros.

Reciprocity formulae for general Dedekind-Rademacher sums

R. R. Hall, J. C. Wilson, D. Zagier (1995)

Acta Arithmetica

Reciprocity formulae for general multiple Dedekind-Rademacher sums and enumeration of lattice points

Abdelmejid Bayad, Abdelaziz Raouj (2010)

Acta Arithmetica

Reciprocity laws for generalized higher dimensional Dedekind sums

Robin Chapman (2000)

Acta Arithmetica

We define a class of generalized Dedekind sums and prove a family of reciprocity laws for them. These sums and laws generalize those of Zagier [6]. The method is based on that of Solomon [5].

Recurrence Formulae for the Coefficients of Modular Forms and Congruences for the Partition Function and for the Coefficients of j(...), (j(...) - 1728)... and (j(...))... .

Torleiv Klove (1968)

Mathematica Scandinavica

Reduction theory using semistability, II.

Daniel R. Grayson (1986)

Commentarii mathematici Helvetici

Refinement of Tate's discriminant bound and non-existence theorems for mod $p$ Galois representations.

Moon, Hyunsuk, Taguchi, Yuichiro (2003)

Documenta Mathematica

Regular trace formula and base change for $G L (n)$

Yuval Z. Flicker (1990)

Annales de l'institut Fourier

The “regular”trace formula, for a test function with a local component which is Iwahori-biinvariant and sufficiently regular with respect to the other components, is developed in the context of a reductive group. It is used to give a simple proof of the theory of base-change for cuspidal automorphic representations of $G L (n)$ which have a supercuspidal component. A purely local proof is given to transfer orbital integrals of sufficiently many spherical functions, by relating them to regular Iwahori functions....

Reguläre Differentialformen des Körpers der Siegelschen Modulfunktionen.

Eberhard Freitag, Klaus Pommerening (1982)

Journal für die reine und angewandte Mathematik

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