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What are the current best known upper bounds on

$$\sum_{n<x}a_n$$

where $a_n$ are defined implicitly by $L(E,s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ where $L(E,s)$ is an Elliptic L Function and $E/\mathbb{Q}$ is an elliptic curve over $\mathbb{Q}$? The notion of "best" might be a bit fuzzy here since on one hand we want a bound that grows slowly with respect to $x$, but on the other we can a bound which goes not get too large as invariants of the curve (such as the conductor) get large.

I would expect something of the type $\left|\sum_{n<x}a_n\right|\ll_E x\log(x)^r$ where $r$ is the rank, judging by the BSD conjecture being initially stated in the form $\prod_{p<x}\frac{N_p}{p}\sim C\log(x)^r$, but the issue is that the BSD conjecture remains a conjecture and there is no immediately obvious way to find an upper bound on $C$ in terms of invariants of the curve.

Any bounds of the quantity $\sum_{n<x}a_n$ or adjacent objects would be highly appreciated. Thank you.

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    $\begingroup$ Are you interested in elliptic curves over $\mathbb Q$? If so, the modularity theorem helps here, unlocking the functional equation, the Rankin-Selberg method, and other tools. $\endgroup$
    – Will Sawin
    Feb 15 at 18:42
  • $\begingroup$ @WillSawin Yes, sorry, I am interested in elliptic curves over $\mathbb{Q}$. Thank you for clarifying. $\endgroup$
    – Milo Moses
    Feb 15 at 18:44
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    $\begingroup$ It's strange to call it "decay" when the sum must increase with $x$. For any cusp form $f = \sum_n a_n q^n$ of weight $2$ we have $a_n \ll_\epsilon n^{1/2 + \epsilon}$ (and I think it's even known that $\sum_{n \leq x} a_n^2 \sim C_f x^2$); if we model the $a_n$ as random numbers of magnitude about $n^{1/2}$ then we expect that $\sum_{n \leq x} a_n \ll_\epsilon n^{3/4 + \epsilon}$, regardless of the rank $-$ though proving such a bound may still be an open problem. $\endgroup$ Feb 15 at 18:57
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    $\begingroup$ @NoamD.Elkies In your random model you must mean $n^{1 + \epsilon}$. My understanding is that such random models will be accurate descriptors of the behavior of these sums for $x$ small relative to the conductor, but bad for $x$ large relative to the conductor, where we can get estimates that are much better than expected for sums of random variables, especially if we use smoothing instead of sharp cutoffs. You are certainly right that this problem may be open - it's a form of RH for the modular L-function... $\endgroup$
    – Will Sawin
    Feb 15 at 19:12
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    $\begingroup$ See also my comment below Will Sawin's response. $\endgroup$
    – GH from MO
    Feb 16 at 15:17
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The Weil bound $a_n \leq d_2(n)\sqrt{n}$ where $n$ is the divisor function gives $ \sum_{n < x} a_n < \sum_{n_1n_2<x} \sqrt{n_1n_2} = O ( x^{3/2} \log x )$. This is the trivial bound in this setting.

To do better than that, the main approach will be to use the modularity theorem, which recognizes $a_n$ as Fourier coefficients of a modular form of weight $2$ and level $N$ equal to the conductor of the elliptic curve.

For $x$ large compared to the conductor $N$, we can do better directly using this modularity, applying the Voronoi summation formula to obtain a bound whose exact form I don't remember off the top of my head but which will have shape $O( x^{1/2 + \epsilon} N^{1/2})$.

You can combine these to get a bound like $O ( x^{1 + \epsilon} N^{1/4})$, which is technically of the form you desire, but whose dependence on $N$ is probably worse than you want.

Doing better than this will be closely related to bounds on the $L$-function of the elliptic curve. In particular, you can do a little better when $x$ is close to $N^{1/2}$ using subconvexity results.

Getting the dependence on $x$ all the way to $ x^{1+\epsilon}$ while keeping the dependence on $N$ small, say $N^{\epsilon}$, is problematic, because it implies

$$L( E, s) = \sum_{n=1}^{\infty} a_n n^{-s} = \sum_{d =2}^{\infty} \left(\sum_{n < d} a_n\right) \left( (d-1)^{-s} - d^{-s} \right)$$ $$ \ll N^\epsilon \sum_{d=2}^{\infty} d \log (d)^r \left( (d-1)^{-s} - d^{-s} \right) ,$$ which, because that sum converges for $s>1$, gives an $O(n^\epsilon)$ estimate of $L(E,s)$ for any $s>1$, which is equivalent to the Lindelöf hypothesis for this $l$-function.

I don't think you will see the rank of the curve playing a big role in this sum. The rank shows up in a sum over primes, which is related to the logarithmic derivative of the $L$-function, much more than it does in this sum, which is related to the $L$-function itself, because zeroes become poles when you take a logarithmic derivative.

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  • $\begingroup$ Very good answer. A classical result in this context is Theorem 4.1 and Remark 5.5 in Chandrasekharan-Narasimhan: Functional equations with multiple gamma factors and the average order of arithmetical functions (1962). One can apply (4.1) for $\lambda_n=\pi n/\sqrt{N}$, $\delta=2$, $A=1$, $u=1/4$, $q=-\infty$. See also Theorem 1 and the subsequent remarks in Hafner-Ivic: On sums of Fourier coefficients of cusp forms (1989). $\endgroup$
    – GH from MO
    Feb 16 at 14:59

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