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Alexandru A. Popa

Publications and source records attributed to Alexandru A. Popa.

15 recordsLinked to original sources
arXiv2026

On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters

In this paper, we establish asymptotic formulas for the first and second twisted moments of $r$-th order Hecke $L$-functions over global fields that contain the $2r$-th roots of unity, for $r\ge 3$. We focus primarily on algebraic number fields. As a consequence, we establish a positive proportion of non-vanishing central values for these $L$-functions, specifically for families of both square-free and $r$-th power-free ideals. Our approach is based on the machinery of multiple Dirichlet series.

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arXiv2026

A decomposition of Weyl group multiple Dirichlet series for symmetrizable Kac-Moody root systems

We study twisted Weyl group multiple Dirichlet series attached to symmetrizable Kac-Moody root systems, using the Chinta-Gunnells method to construct their $p$-parts. Our main result is a decomposition theorem for functions invariant under the twisted Chinta-Gunnells action: under natural analytic hypotheses, such a function has a unique expansion in terms of shifted Chinta-Gunnells averages, indexed by the dominant weights in the highest weight module determined by the twisting parameter. In particular, we show that this decomposition holds for twisted multiple Dirichlet series over rational function fields. For finite root systems, these results were proved by Friedlander. We also show that the relevant Chinta-Gunnells averages admit analytic continuation to the interior of the complexified Tits cone. In the affine $\widetilde{A}_1$ case, we prove extra functional equations, not arising from the Weyl group, for the untwisted average and for averages twisted by fundamental weights. As a consequence, we obtain an explicit formula for the multiple Dirichlet series with square-free twisting parameters, and show that it also satisfies an extra functional equation.

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arXiv2023

Quadratic Weyl group multiple Dirichlet series of Type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$

In this paper and its sequel \cite{DPP}, we investigate the precise relationship between the quadratic affine Weyl group multiple Dirichlet series in the sense of \cite{CG1, BD}, and those defined axiomatically by Whitehead \cite{White2} and \cite{White1}. In particular, we show that the axiomatic quadratic Weyl group multiple Dirichlet series of type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$ over rational function fields of odd characteristic admits meromorphic continuation to the interior of the corresponding complexified Tits cone. We shall also determine the polar divisor of this function, and compute the residue at each of its poles. As a consequence, we obtain an \emph{exact} formula for a weighted 4-th moment of quadratic Dirichlet $L$-functions over rational function fields; we shall also derive an asymptotic formula for this weighted moment that is expected to generalize to any global field.

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arXiv2023

Residues of quadratic Weyl group multiple Dirichlet series

We give explicit formulas for the residue of the Chinta-Gunnells average attached to a finite irreducible root system, at the polar divisor corresponding to a simple short root. The formula describes the residue in terms of the average attached to the root subsystem orthogonal to the relevant simple root. As a consequence, we obtain similar formulas for the residues of quadratic Weyl group multiple Dirichlet series over the rational function field and over the Gaussian field. The residue formula also allows us to obtain a new expression for the Chinta-Gunnells average of a finite irreducible root system, as an average over a maximal parabolic subgroup of a rational function that has an explicit description reflecting the combinatorics of the root system.

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arXiv2023

A trace formula for Hecke operators on Fuchsian groups

In this paper we give a trace formula for Hecke operators acting on the cohomology of a Fuchsian group of finite covolume, with coefficients in a module $V$. The proof is based on constructing an operator whose trace on $V$ equals the Lefschetz number of the Hecke correspondence on cohomology, generalizing the operator introduced together with Don Zagier for the modular group.

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arXiv2018

A generalization of Ramanujan's congruence to modular forms of prime level

We prove congruences between cuspidal newforms and Eisenstein series of prime level, which generalize Ramanujan's congruence. Such congruences were recently found by Billerey and Menares, and we refine them by specifying the Atkin-Lehner eigenvalue of the newform involved. We show that similar refinements hold for the level raising congruences between cuspidal newforms of different levels, due to Ribet and Diamond. The proof relies on studying the new subspace and the Eisenstein subspace of the space of period polynomials for the congruence subgroup $Γ_0(N)$, and on a version of Ihara's lemma.

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arXiv2017

On the trace formula for Hecke operators on congruence subgroups

We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of modular forms, and it generalizes an approach developed by Don Zagier and the author for the modular group. This approach leads to a very simple formula for the trace on the space of cusp forms plus the trace on the space of modular forms. As applications, we investigate what happens when varying the weight or the level in the trace formula.

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arXiv2017

On the trace formula for Hecke operators on congruence subgroups, II

In a previous paper, we obtained a general trace formula for double coset operators acting on modular forms for congruence subgroups, expressed as a sum over conjugacy classes. Here we specialize it to the congruence subgroups $Γ_0(N)$ and $Γ_1(N)$, obtaining explicit formulas in terms of class numbers for the trace of a composition of Hecke and Atkin-Lehner operators. The formulas are among the simplest in the literature, and hold without any restriction on the index of the operators. We give two applications of the trace formula for $Γ_1(N)$: we determine explicit trace forms for $Γ_0(4)$ with Nebentypus, and we compute the limit of the trace of a fixed Hecke operator as the level $N$ tends to infinity.

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arXiv2014

Pair correlation of hyperbolic lattice angles

Let $ω$ be a point in the upper half plane, and let $Γ$ be a discrete, finite covolume subgroup of $\mathrm{PSL}_2(\mathbb{R})$. We conjecture an explicit formula for the pair correlation of the angles between geodesic rays of the lattice $Γω$, intersected with increasingly large balls centered at $ω$. We prove this conjecture for $Γ=\mathrm{PSL}_2(\mathbb{Z})$ and $ω$ an elliptic point.

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arXiv2013

Pair correlation of angles between reciprocal geodesics on the modular surface

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as well as arithmetic information about the associated reciprocal classes of binary quadratic forms. One striking feature is the absence of a gap beyond zero in the limiting distribution, contrasting with the analog Euclidean situation.

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arXiv2013

On the Petersson scalar product of arbitrary modular forms

We consider a natural extension of the Petersson scalar product to the entire space of modular forms of integral weight $k\ge 2$ for a finite index subgroup of the modular group. We show that Hecke operators have the same adjoints with respect to this inner product as for cusp forms, and we show that the Petersson product is nondegenerate for $Γ_1(N)$ and $k>2$. For $k=2$ we give examples when it is degenerate, and when it is nondegenerate.

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arXiv2013

An algebraic property of Hecke operators and two indefinite theta series

We prove an algebraic property of the elements defining Hecke operators on period polynomials associated with modular forms, which implies that the pairing on period polynomials corresponding to the Petersson scalar product of modular forms is Hecke equivariant. As a consequence of this proof, we derive two indefinite theta series identities which can be seen as analogues of Jacobi's formula for the theta series associated with the sum of four squares.

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arXiv2013

Modular forms and period polynomials

We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for $Γ_1(N)$: an extension of the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficients Theorem of Manin.

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