Heegner divisors,L-functions and harmonic weak Maass forms
نویسندگان
چکیده
منابع مشابه
Heegner Divisors, L-functions and Harmonic Weak Maass Forms
Recent works, mostly related to Ramanujan’s mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as “generating functions” for central values and derivatives of quadratic twists of weight 2 modular L-functions. To obtain these results, we cons...
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Arakelov geometry, which is a mixture of algebraic geometry at finite primes (of a number field) and real analysis at infinite primes, was invented by Arakelov [Ar] in 70s to ‘compactify’ an arithmetic variety (see also [Fa2]). It has become a very important part of modern number theory after Faltings’ proof of the Mordell conjecture (see for example [Fa1], [So]) and the celebrated Gross-Zagier...
متن کاملDifferential Operators and Harmonic Weak Maass Forms
For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2−k (resp. D) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms have transcendental coefficients, we show that those...
متن کاملComputation of Harmonic Weak Maass Forms
Harmonic weak Maass forms of half-integral weight are the subject of many recent works. They are closely related to Ramanujan’s mock theta functions, their theta lifts give rise to Arakelov Green functions, and their coefficients are often related to central values and derivatives of Hecke L-functions. We present an algorithm to compute harmonic weak Maass forms numerically, based on the automo...
متن کاملArithmetic Properties of Non-harmonic Weak Maass Forms
The arithmetic behavior of the partition function has been of great interest. For example, we have the famous Ramanujan congruences p(5n+ 4) ≡ 0 (mod 5), p(7n+ 5) ≡ 0 (mod 7), p(11n+ 6) ≡ 0 (mod 11) for every n ≥ 0. In a celebrated paper Ono [13] treated this type of congruence systematically. Combining Shimura’s theory of modular forms of half-integral weight with results of Serre on modular f...
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ژورنال
عنوان ژورنال: Annals of Mathematics
سال: 2010
ISSN: 0003-486X
DOI: 10.4007/annals.2010.172.2135