Moments of characteristic polynomials and their derivatives for $SO(2N)$ and $USp(2N)$ and their application to one-level density in families of elliptic curve $L$-functions
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arXiv
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| Natura: | Preprint |
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2024
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| author | Cooper, I. A. Snaith, N. C. |
| author_facet | Cooper, I. A. Snaith, N. C. |
| contents | Using the ratios theorems, we calculate the leading order terms in $N$ for the following averages of the characteristic polynomial and its derivative: $\left< \left|Λ_A(1 )\right| ^{r} \frac{ Λ_A'(\mathrm{e}^{\mathrm{i} ϕ}) }{ Λ_A(\mathrm{e}^{\mathrm{i} ϕ})} \right>_{SO(2N)}$ and $\left< \left|Λ_A(1 )\right| ^{r} \frac{ Λ_A'(\mathrm{e}^{\mathrm{i} ϕ}) }{ Λ_A(\mathrm{e}^{\mathrm{i} ϕ})} \right>_{USp(2N)}$. Our expression, derived for integer $r$, permits analytic continuation in $r$ and we conjecture that this agrees with the above averages for non-integer exponents. We use this result to obtain an expression for the one level density of the `excised ensemble', a subensemble of $SO(2N)$, to next-to-leading order in $N$. We then present the analogous calculation for the one level density of quadratic twists of elliptic curve $L$-functions, taking into account a number theoretical bound on the central values of the $L$-functions. The method we use to calculate the above random matrix averages uses the contour integral form of the ratios theorems, which are a key tool in the growing literature on averages of characteristic polynomials and their derivatives, and as we evaluate the next-to-leading term for large matrix size $N$, this leads to some multi-dimensional contour integrals that are slightly asymmetric in the integration variables, which might be useful in other work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02024 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moments of characteristic polynomials and their derivatives for $SO(2N)$ and $USp(2N)$ and their application to one-level density in families of elliptic curve $L$-functions Cooper, I. A. Snaith, N. C. Mathematical Physics Number Theory 11M50 (Primary) 60B20 (Secondary) Using the ratios theorems, we calculate the leading order terms in $N$ for the following averages of the characteristic polynomial and its derivative: $\left< \left|Λ_A(1 )\right| ^{r} \frac{ Λ_A'(\mathrm{e}^{\mathrm{i} ϕ}) }{ Λ_A(\mathrm{e}^{\mathrm{i} ϕ})} \right>_{SO(2N)}$ and $\left< \left|Λ_A(1 )\right| ^{r} \frac{ Λ_A'(\mathrm{e}^{\mathrm{i} ϕ}) }{ Λ_A(\mathrm{e}^{\mathrm{i} ϕ})} \right>_{USp(2N)}$. Our expression, derived for integer $r$, permits analytic continuation in $r$ and we conjecture that this agrees with the above averages for non-integer exponents. We use this result to obtain an expression for the one level density of the `excised ensemble', a subensemble of $SO(2N)$, to next-to-leading order in $N$. We then present the analogous calculation for the one level density of quadratic twists of elliptic curve $L$-functions, taking into account a number theoretical bound on the central values of the $L$-functions. The method we use to calculate the above random matrix averages uses the contour integral form of the ratios theorems, which are a key tool in the growing literature on averages of characteristic polynomials and their derivatives, and as we evaluate the next-to-leading term for large matrix size $N$, this leads to some multi-dimensional contour integrals that are slightly asymmetric in the integration variables, which might be useful in other work. |
| title | Moments of characteristic polynomials and their derivatives for $SO(2N)$ and $USp(2N)$ and their application to one-level density in families of elliptic curve $L$-functions |
| topic | Mathematical Physics Number Theory 11M50 (Primary) 60B20 (Secondary) |
| url | https://arxiv.org/abs/2409.02024 |