Quelques mathématiques de la cryptographie à clés publiques

Let $r(N)$ be the number of representations of $N$ as a sum of two squares. For every integer $n\ge 2$ we prove an asymptotic formula for $\sum_{x\equiv e \bmod 2}Φ(x/M)\,r(n^2+x^2)$, that is, for the number of integral solutions of $x_1^2+x_2^2-x_3^2=n^2$ with $x_3\equiv e \bmod 2$ in a range $|x_3|\ll M$, with smooth weight $Φ$. The formula has an explicit secondary term and a power saving in the error term, and it is non-trivial as soon as $M\ge n^{5/6+\varepsilon}$, that is, $M\ge D^{1/2-1/12+\varepsilon}$ with $D=n^2$. The leading term is $\frac{4}π(\log n)\intΦ(x/M)\,dx$, independently of the arithmetic of $n$; this corrects the main term stated in the published version (Ramanujan J. 63 (2024), 293-337). The proof factorises $n+ix_3$ in $\mathbb{Z}[i]$, which leads to Kloosterman sums. These are estimated by Weil's bound, combined with a Plancherel argument that exploits the curvature of the underlying quadric. This paper was written in collaboration with Claude (Opus 5.5, High), an AI model developed by Anthropic.

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arXiv2026

Bessel Periods on $U(2,1) \times U(1,1)$, Relative Trace Formula and Non-Vanishing of Central $L$-values

In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations $(π, π')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over $π$). Using these, we obtain quantitative non-vanishing results for the Rankin-Selberg central $L$-values $L(1/2, π\times π')$, which are of degree twelve over ${\mathbb Q}$, with concomitant difficulty in applying standard methods, especially since we are in a `conductor dropping' situation. We use the relative trace formula, and the orbital integrals are evaluated rather than compared with others. Besides their intrinsic interest, non-vanishing of these critical values also leads, by known results, to deducing certain associated Selmer groups have rank zero.

math.NT↗

arXiv2026

Rational points on $X_0(N)^*$ when $N$ is non-squarefree

Let $N$ be a positive integer and $X_0(N)^*$ be the quotient of the modular curve $X_0(N)$ by its full group of Atkin-Lehner involutions. Elkies conjectures that the rational points on $X_0(N)^*$ are only cusps or CM points when $N$ is large enough. We establish an integrality result for the $j$-invariants of non-cuspidal rational points on $X_0(N)^*$ when $N$ is non-squarefree, representing a significant step toward resolving a key subcase of Elkies' conjecture. To this end, we prove the existence of rank-zero quotients of certain modular Jacobians $J_0(pq)$. Furthermore, in the non-squarefree case we provide a complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$, when they are finite. In the process we identify exceptional rational points on $X_0(75)^*$ and $X_0(147)^*$ which were not known before.

math.NT↗