Is hasse bound of elliptic curve larger when we discard smoothness?

Question

Let $E$ be an elliptic curve over finite field $\Bbb{F}_q$.

The inequality $\mid 1+q-\sharp E(\Bbb{F}_q)\mid\leq 2\sqrt{q}$・・・① is well known.

My question is, does this inequality still hold if we discard smoothness of $E$, in other words, does this inequality hold for singular genus $1$ curves?

The proof of this theorem seems invalid if we discard smoothness, but I'm stucking with finding counterexamples　of ①, that is, I want to know an example of singular genus $1$ curve $C$ and prime $p$ which satisfies $\mid 1+p-\sharp C(\Bbb{F}_p)\mid > 2\sqrt{p}$.

Answer 1

After a few experimental lines with sage, that makes it easy to count points on curves, i found the following example in characteristic $p=5$. Consider $C$ to be the curve given in homogeneous coordinates by: $$ \bbox[yellow]{\qquad x^2(x-y)(x-2y)=z^4 \qquad\text{ over }\Bbb F_5\ . \qquad} $$ It has the following $11$ rational points: $$ \begin{aligned} & (0 : 1 : 0) \\ & (1 : 0 : 1) \\ & (1 : 1 : 0) \\ & (1 : 4 : 1) \\ & (2 : 0 : 1) \\ & (2 : 1 : 0) \\ & (2 : 3 : 1) \\ & (3 : 0 : 1) \\ & (3 : 2 : 1) \\ & (4 : 0 : 1) \\ & (4 : 1 : 1) \end{aligned} $$

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Further examples of the same shape, $$ \bbox[yellow]{\qquad x^2(x-y)(x-ay) = z^4\qquad\text{ over }\Bbb F_p\ ,\text{ $p$ prime,} \qquad} $$ with a small $p$ (up to $101$) are in the following table: $$ \begin{array}{|r|r|r|r|} \hline p & a & \#C(\Bbb F_p) & p + 1 - 2\sqrt p & p + 1 + 2\sqrt p\\\hline 5 & 2 & 11 & 1.528\dots & 10.472\dots\\ 5 & 3 & 11 & 1.528\dots & 10.472\dots\\\hline 17 & 3 & 27 & 9.754\dots & 26.246\dots\\ 17 & 5 & 27 & 9.754\dots & 26.246\dots\\ 17 & 6 & 27 & 9.754\dots & 26.246\dots\\ 17 & 7 & 27 & 9.754\dots & 26.246\dots\\ 17 & 11 & 27 & 9.754\dots & 26.246\dots\\ 17 & 14 & 27 & 9.754\dots & 26.246\dots\\\hline 29 & 5 & 19 & 19.230\dots & 40.770\dots\\ 29 & 6 & 19 & 19.230\dots & 40.770\dots\\ 29 & 16 & 19 & 19.230\dots & 40.770\dots\\ 29 & 20 & 19 & 19.230\dots & 40.770\dots\\\hline 37 & 5 & 51 & 25.834\dots & 50.166\dots\\ 37 & 8 & 51 & 25.834\dots & 50.166\dots\\ 37 & 13 & 51 & 25.834\dots & 50.166\dots\\ 37 & 14 & 51 & 25.834\dots & 50.166\dots\\ 37 & 15 & 51 & 25.834\dots & 50.166\dots\\ 37 & 17 & 51 & 25.834\dots & 50.166\dots\\ 37 & 18 & 51 & 25.834\dots & 50.166\dots\\ 37 & 20 & 51 & 25.834\dots & 50.166\dots\\ 37 & 22 & 51 & 25.834\dots & 50.166\dots\\ 37 & 24 & 51 & 25.834\dots & 50.166\dots\\ 37 & 32 & 51 & 25.834\dots & 50.166\dots\\ 37 & 35 & 51 & 25.834\dots & 50.166\dots\\\hline 53 & 4 & 39 & 39.440\dots & 68.560\dots\\ 53 & 6 & 39 & 39.440\dots & 68.560\dots\\ 53 & 9 & 39 & 39.440\dots & 68.560\dots\\ 53 & 10 & 39 & 39.440\dots & 68.560\dots\\ 53 & 16 & 39 & 39.440\dots & 68.560\dots\\ 53 & 40 & 39 & 39.440\dots & 68.560\dots\\ 53 & 44 & 39 & 39.440\dots & 68.560\dots\\ 53 & 47 & 39 & 39.440\dots & 68.560\dots\\ 53 & 52 & 39 & 39.440\dots & 68.560\dots\\\hline 101 & 2 & 123 & 81.900\dots & 122.100\dots\\ 101 & 7 & 123 & 81.900\dots & 122.100\dots\\ 101 & 8 & 123 & 81.900\dots & 122.100\dots\\ 101 & 10 & 123 & 81.900\dots & 122.100\dots\\ \end{array} $$ ... and so on. (For $p=101$ there are many other counterexamples...)

As seen, not too many primes occur in the list, those close to squares (and bigger then them) are favored to hit the upper bound. Searching for curves with the same pattern, and delivering only the first curve for each prime only, we have the following table:

$$ \begin{array}{|r|r|r|r|} \hline p & a & \#C(\Bbb F_p) & p + 1 - 2\sqrt p & p + 1 + 2\sqrt p\\\hline 5 & 2 & 11 & 1.528\dots & 10.472\dots\\ 17 & 3 & 27 & 9.754\dots & 26.246\dots\\ 29 & 5 & 19 & 19.230\dots & 40.770\dots\\ 37 & 5 & 51 & 25.834\dots & 50.166\dots\\ 53 & 4 & 39 & 39.440\dots & 68.560\dots\\ 101 & 2 & 123 & 81.900\dots & 122.100\dots\\ 109 & 2 & 131 & 89.119\dots & 130.881\dots\\ 173 & 6 & 147 & 147.694\dots & 200.306\dots\\ 197 & 3 & 227 & 169.929\dots & 226.071\dots\\ 229 & 11 & 199 & 199.735\dots & 260.265\dots\\ 257 & 3 & 291 & 225.938\dots & 290.062\dots\\ 293 & 10 & 259 & 259.766\dots & 328.234\dots\\ 401 & 6 & 443 & 361.950\dots & 442.050\dots\\ 409 & 11 & 451 & 369.553\dots & 450.447\dots\\ 457 & 4 & 415 & 415.245\dots & 500.755\dots\\ 577 & 5 & 627 & 529.958\dots & 626.042\dots\\ 641 & 7 & 591 & 591.364\dots & 692.636\dots\\ 677 & 2 & 731 & 625.962\dots & 730.038\dots\\ 701 & 2 & 755 & 649.047\dots & 754.953\dots\\ 733 & 10 & 679 & 679.852\dots & 788.148\dots\\ 809 & 6 & 867 & 753.114\dots & 866.886\dots\\ 857 & 8 & 799 & 799.451\dots & 916.549\dots\\ 977 & 2 & 915 & 915.486\dots & 1040.514\dots\\ \hline \end{array} $$

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Used code (printing the results to be plugged in into the mathjax array blocks):

for p in primes(1000):
    F = GF(p)
    R.<x,y,z> = PolynomialRing(F)
    bound0  = float(p + 1 - 2*sqrt(p))
    bound1  = float(p + 1 + 2*sqrt(p))
    for a in F:
        C = Curve( x^2*(x - y)*(x - a*y) - z^4)
        if C.genus() != 1:
            continue
        points = C.rational_points()
        if abs(p + 1 - len(points))^2 > 4*p:
            print(f'{p} & {a} & {len(points)} & '
                  f'{bound0:.3f}\\dots & {bound1:.3f}\\dots\\\\')
            # break # decomment to see only first curve for the present p 

Answer 2

There is the paper "A Weil Theorem for Singular Curves" by Y Aubry and M Perret in the monograph Arithmetic Geometry and Coding Theory, Pelikaan et al, de Gruyter, 1996.

We generalize Weil's theorem on the number of rational points of smooth curves over a finite field to singular ones

If you google by title you can find a scan of the paper. I am not an expert in this but it seems that $$ | 1+q-\#E(\mathbb{F}_q) | \leq 2 g \sqrt{q}+\Delta_E\leq 2 g \sqrt{q}+\pi-g\leq2 \pi \sqrt{g} $$

where $\pi$ is the arithmetic genus and $g$ the geometric genus of the curve, $$ \Delta_x=\#\{\tilde{E}(\mathbb{F}_q)\setminus E(\mathbb{F}_q)\}, $$ and $\tilde{E}$ is the normalization of the curve $E$ with an explicit product of cyclotomic polynomials.

Also, Bach, E. (1996). "Weil bounds for singular curves", Applicable Algebra in Engineering, Communication and Computing 7(4), 289–298. doi:10.1007/bf01195534