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Nikolaos Diamantis

Publications and source records attributed to Nikolaos Diamantis.

29 records · Page 2Linked to original sources
arXiv2016

Regularized inner products and errors of modularity

We develop a regularization for Petersson inner products of arbitrary weakly holomorphic modular forms, generalizing several known regularizations. As one application, we extend work of Duke, Imamoglu, and Toth on regularized inner products of weakly holomorphic modular forms of weights $0$ and $3/2$. These regularized inner products can be evaluated in terms of the coefficients of holomorphic parts of harmonic Maass forms of dual weights. Moreover, we study the errors of modularity of the holomorphic parts of such a harmonic Maass forms and show that they induce cocyles in the first parabolic cohomology group introduced by Bruggeman, Choie, and the second author. This provides explicit representatives of the cohomology classes constructed abstractly and in a very general setting in their work.

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arXiv2014

Holomorphic automorphic forms and cohomology

We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. Knopp generalized this to real weights. We show that for weights that are not an integer at least 2 the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. We impose no condition on the growth of the automorphic forms at the cusps. For real weights that are not an integer at least 2 we similarly characterize the space of cusp forms and the space of entire automorphic forms. We give a relation between the cohomology classes attached to holomorphic automorphic forms of real weight and the existence of harmonic lifts. A tool in establishing these results is the relation to cohomology groups with values in modules of "analytic boundary germs", which are represented by harmonic functions on subsets of the upper half-plane. Even for positive integral weights cohomology with these coefficients can distinguish all holomorphic automorphic forms, unlike the classical Eichler theory.

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arXiv2011

Kernels for products of L-functions

The Rankin-Cohen bracket of two Eisenstein series provides a kernel yielding products of the periods of Hecke eigenforms at critical values. Extending this idea leads to a new type of Eisenstein series built with a double sum. We develop the properties of these series and their non-holomorphic analogs and show their connection to values of L-functions outside the critical strip.

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arXiv2011

Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions

We introduce a new technique of completion for 1-cohomology which parallels the corresponding technique in the theory of mock modular forms. This technique is applied in the context of non-critical values of L-functions of GL(2,Q) cusp forms. We prove that a generating series of non-critical values can be interpreted as a mock period function we define in analogy with period polynomials. Further, we prove that non-critical values can be encoded into a sesquiharmonic Maass form. Finally, we formulate and prove an Eichler-Shimura-type isomorphism for the space of mock period functions.

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arXiv2010

Higher order Maass forms

We determine the size of spaces of higher order Maass forms of even weight for cofinite discrete subgroups of PSL(2,R) with cusps. If exponential growth at the cusps is allowed, the spaces of Maass forms of a given order are as large as algebraic constrictions allow. We show the analogous statement for the spaces of holomorphic forms. For functions on the universal covering group of PSL(2,R) we introduce the concept of generalized weight. For the resulting spaces of higher order Maass forms with even generalized weight we show that the size is maximal.

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arXiv2009

Kernels of $L$-functions of cusp forms

We give a new expression for the inner product of two kernel functions associated to a cusp form. Among other applications, it yields an extension of a formula of Kohnen and Zagier, and another proof of Manin's Periods Theorem. Cohen's representation of these kernels as series is also generalized.

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arXiv2009

New percolation crossing formulas and second-order modular forms

We consider the three new crossing probabilities for percolation recently found via conformal field theory by Simmons, Kleban and Ziff. We prove that all three of them (i) may be simply expressed in terms of Cardy's and Watts' crossing probabilities, (ii) are (weakly holomorphic) second-order modular forms of weight 0 (and a single particular type) on the congruence group $Γ(2)$, and (iii) under some technical assumptions (similar to those used by Kleban and Zagier, are completely determined by their transformation laws. The only physical input in (iii) is Cardy's crossing formula, which suggests an unknown connection between all crossing-type formulas.

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arXiv2008

Iterated Integrals and higher order automorphic forms

Higher order automorphic forms have recently been introduced to study important questions in number theory and mathematical physics. We investigate the connection between these functions and Chen's iterated integrals. Then using Chen's theory, we prove a structure theorem for automorphic forms of all orders. This allows us to define an analogue of a mixed Hodge structure on a space of higher order automorphic forms.

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