Ramanujan's class invariants, Kronecker's limit formula and modular equations (III)
Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary.
Rank zero quadratic twists of modular elliptic curves
Rank-Crank type PDE's for higher level Appell functions
Rankin–Cohen brackets and representations of conformal Lie groups
This is an extended version of a lecture given by the author at the summer school “Quasimodular forms and applications” held in Besse in June 2010.The main purpose of this work is to present Rankin-Cohen brackets through the theory of unitary representations of conformal Lie groups and explain recent results on their analogues for Lie groups of higher rank. Various identities verified by such covariant bi-differential operators will be explained by the associativity of a non-commutative product...
Rational period functions and indefinite binary quadratics forms. I.
Rational points on modular curves and abelian varieties.
Rational points on the modular curves
Real zeros of holomorphic Hecke cusp forms
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
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.
Recurrence Formulae for the Coefficients of Modular Forms and Congruences for the Partition Function and for the Coefficients of j(...), (j(...) - 1728)... and (j(...))... .
Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids.
Représentations -adiques
Representations of squares by certain quinary quadratic forms
Residually reductible representations and modular forms