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Register a new userWide moments of $L$-functions I: Twists by class group characters of imaginary quadratic fields
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Asbjorn Christian Nordentoft
Publication date
2021
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English
Authors
Asbjorn Christian Nordentoft
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We calculate certain wide moments of central values of Rankin--Selberg $L$-functions $L(piotimes Omega, 1/2)$ where $pi$ is a cuspidal automorphic representation of $mathrm{GL}_2$ over $mathbb{Q}$ and $Omega$ is a Hecke character (of conductor $1$) of an imaginary quadratic field. This moment calculation is applied to obtain weak simultaneous non-vanishing results, which are non-vanishing results for different Rankin--Selberg $L$-functions where the product of the twists is trivial. The proof relies on relating the wide moments to the usual moments of automorphic forms evaluated at Heegner points using Waldspurgers formula. To achieve this, a classical version of Waldspurgers formula for general weight automorphic forms is proven, which might be of independent interest. A key input is equidistribution of Heegner points (with explicit error-terms) together with non-vanishing results for certain period integrals. In particular, we develop a soft technique for obtaining non-vanishing of triple convolution $L$-functions.
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We obtain the asymptotic formula with an error term $O(X^{frac{1}{2} + varepsilon})$ for the smoothed first moment of quadratic twists of modular $L$-functions. We also give a similar result for the smoothed first moment of the first derivative of quadratic twists of modular $L$-functions. The argument is largely based on Youngs recursive method [19,20].
We give an explicit construct of a harmonic weak Maass form $F_{Theta}$ that is a lift of $Theta^3$, where $Theta$ is the classical Jacobi theta function. Just as the Fourier coefficients of $Theta^3$ are related to class numbers of imaginary quadratic fields, the Fourier coefficients of the holomorphic part of $F_{Theta}$ are associated to class numbers of real quadratic fields.
Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler--Shimura isomorphism and contain information about automorphic $L$-functions. In this paper we prove that central values of additive twists of the $L$-function associated to a holomorphic cusp form $f$ of even weight $k$ are asymptotically normally distributed. This generalizes (to $kgeq 4$) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore we give as an application an asymptotic formula for the averages of certain wide families of automorphic $L$-functions, consisting of central values of the form $L(fotimes chi,1/2)$ with $chi$ a Dirichlet character.
We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of $q$, and we use the theory of Deligne and Katz to estimate certain complete exponential sums in several variables and prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of $q$. When at least one of the forms $f$ and $g$ is non-cuspidal, we obtain an asymptotic formula for the mixed second moment of twisted $L$-functions with a power saving error term. In particular, when both are non-cuspidal, this gives a significant improvement on M.~Youngs asymptotic evaluation of the fourth moment of Dirichlet $L$-functions. In the general case, the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in Kloosterman sums.
The Mordell-Weil groups $E(mathbb{Q})$ of elliptic curves influence the structures of their quadratic twists $E_{-D}(mathbb{Q})$ and the ideal class groups $mathrm{CL}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) in mathbb{Z}^2$, we define a family of homomorphisms $Phi_{u,v}: E(mathbb{Q}) rightarrow mathrm{CL}(-D)$ for particular negative fundamental discriminants $-D:=-D_E(u,v)$, which we use to simultaneously address questions related to lower bounds for class numbers, the structures of class groups, and ranks of quadratic twists. Specifically, given an elliptic curve $E$ of rank $r$, let $Psi_E$ be the set of suitable fundamental discriminants $-D<0$ satisfying the following three conditions: the quadratic twist $E_{-D}$ has rank at least 1; $E_{text{tor}}(mathbb{Q})$ is a subgroup of $mathrm{CL}(-D)$; and $h(-D)$ satisfies an effective lower bound which grows asymptotically like $c(E) log (D)^{frac{r}{2}}$ as $D to infty$. Then for any $varepsilon > 0$, we show that as $X to infty$, we have $$#, left{-X < -D < 0: -D in Psi_E right } , gg_{varepsilon} X^{frac{1}{2}-varepsilon}.$$ In particular, if $ell in {3,5,7}$ and $ell mid |E_{mathrm{tor}}(mathbb{Q})|$, then the number of such discriminants $-D$ for which $ell mid h(-D)$ is $gg_{varepsilon} X^{frac{1}{2}-varepsilon}.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2.
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