Papers

We prove that every number field of degree $n$ and absolute discriminant $D$ has a defining integral polynomial of degree $n$ whose coefficients are all $O_n(\sqrt D)$ in size, thus resolving a 1998 conjecture of Ruppert. Our proof compares ideal lattices with their trace duals, via which we handle all number fields with the exception of CM fields whose maximal totally real subfields have small discriminant. We then treat the remaining fields by averaging completed Epstein zeta functions over all possible ideal classes and archimedean scalings, for which we make critical use of Hecke's formula and the subconvexity theorem of Michel and~Venkatesh. Our proof does not use the Quasi-Riemann Hypothesis.

We prove the full generating series arithmetic Siegel–Weil formula on the product of two modular curves, with arbitrarily singular coefficients. This is the case $\operatorname{GSpin}(2,2)$ in Kudla's program.

Building on the large sieve, we prove a new sieve estimate for sets which are missing various residue classes modulo prime squares. For certain interesting residue class configurations, our bound is a significant improvement over the corresponding Johnsen-Selberg sieve inequality. Our application of this sieve estimate is motivated by Ostmann's problem. We study additive decompositions of sums of squares and show that any non-trivial sumset decomposition must consist of two sets of roughly equal size: If $A_1, A_2 \subseteq \mathbb{Z}$ with $A_1 + A_2 = \{x^2 + y^2 : x, y \in \mathbb{Z}\}$ and $|A_1|, |A_2| \geq 2$, then $\sqrt{x}/(\log x)^{9/2} \ll |A_i \cap [1, x]| \ll \sqrt{x}(\log x)^4$ for $i \in \{1, 2\}$.

For a fixed matrix-polynomial trajectory in $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$, we give a necessary and sufficient intrinsic Diophantine condition for its time averages to equidistribute at a power rate on its homogeneous orbit closure. The condition uses a lattice in the space of polynomial functions on the normalized orbit-closure group, each admitting a representative of total degree at most two in the matrix entries. These polynomial tests have a common degree bound of two, independent of the curve degree and optimal in general. The proof begins with a classification of the possible orbit-closure groups and derives lower bounds for polynomial coefficient vectors from the intrinsic condition. It combines quantitative nilmanifold equidistribution, unipotent tangent approximation, and effective flow estimates, with arithmetic separation and quantitative nondivergence controlling cusp excursions and periodic-orbit obstructions.

We study the category of constructible $p$-Kummer-étale sheaves on the log adic generic fiber $(\mathrm{X},\mathcal{M}_{\mathrm{D}})_\eta$ for the stratification induced by a horizontal normal crossings divisor $\mathrm{D}$, showing that it is antiequivalent to a category of finitely generated unit Laurent crystals on the log prismatic site. We give a local description of finitely generated unit crystals on a framed log affine in terms of $(\varphi,\Gamma)$-modules, and show pushforward compatibility for strict log smooth proper morphisms.

We study the behaviour of partially de Rham Selmer groups of $B$-pairs along abelian $p$-adic Lie extensions. In the cyclotomic case we compare them with the universal norms studied by Berger. For analytic Lubin–Tate $(\varphi,\Gamma)$-modules we prove that the $\Psi=1$ eigenspace is coadmissible of the expected rank over the algebra of analytic distributions and establish a comparison result between its Lie-algebra coinvariants and the projective limit of the partially de Rham Selmer groups of the attached $B$-pair.

We prove two height inequalities for rational points on abelian varieties over function fields of characteristic 0. The first is the GCD inequality, which shows that the height with respect to a closed subscheme of an abelian variety of codimension at least 2 is much smaller than the usual height function. This can be viewed as a compact and geometric analogue of a theorem of Levin. The second is a special case of the (geometric) abc conjecture proposed by Vojta, which compares the truncated counting function and the usual counting function. This is a compact analogue of a theorem of Guo, Nguyen, Sun and Wang, and also resolves a conjecture of Buium, which was only known for traceless abelian varieties or constant abelian varieties previously. The proofs of these inequalities follow a common three-step pattern: from microscopic local estimates to local estimates, and then to global estimates. Building on the canonical Xie-Yuan heights, we control discrete counting functions by continuous integrals of Betti forms. The main new tools are uniform Nevanlinna-type estimates for families of linear entire curves on abelian varieties; combined with covering arguments based on Besicovitch theory and averaging, these handle overlaps and gaps precisely.

Shute showed that the leading constant in the asymptotic formula for squareful values of the form $ax^2+by^2$ does not match with the conjecture established by Pieropan, Smeets, Tanimoto and Várilly-Alvarado. In this paper, $S$-squareful values of this form are considered for $S$ the set of primes that divide $a$ and $b$. We prove an asymptotic formula for these numbers and show that the leading constant of the asymptotic formula still does not match the conjecture. Furthermore, we consider the updated conjecture established by Chow, Loughran, Takloo-Bighash, and Tanimoto regarding the leading constant for Campana points in the given setting and show that this constant is consistent with our computed constant.

For each positive integer $N$, let $p_i$ be the $i$-th prime for $i\in\{1,\ldots,N\}$, and set $$ \Phi_N=\left\{\prod_{i=1}^{N}p_i^{a_i}:a_i\in\{0,1,\ldots,N-1\}\right\}. $$ Write $f(N)$ for the largest cardinality of a subset of $\Phi_N$ containing no two distinct legs of an integral right triangle. We prove $$ N^{N-1}(\log\log\log N)\ll f(N). $$

Let $E/K$ be an elliptic curve over a field defined by the short Weierstrass equation and let $\mathbf{P}=(P_1, P_2) \in E(K)^2\setminus \{(O, O), (\pm P, \pm P)\}$ be a pair of non-torsion points, where $O$ denotes the point at infinity. We give a uniform upper bound for the quantity $\min(\nu(\Phi_{(v_1, v_2)}(\mathbf{P})), 2\nu(\Psi_{(v_1, v_2)}(\mathbf{P})))$, where $\nu$ is a normalised discrete valuation associated to primes where $E$ has bad reduction and $\Psi^2_{(v_1, v_2)}(\mathbf{P})$, $\Phi_{(v_1, v_2)}(\mathbf{P})$ are rational functions for describing $\mathbb{Z}$–linear combination of points $[v_1]P_1+[v_2]P_2$ on $E$, higher–dimensional analogues of division polynomials $\psi^2_n, \phi_n$ known as net polynomials. They are also elliptic nets, which are $n$–dimensional arrays with values in $K$ satisfying the same nonlinear recurrence relation that division polynomials do as well. Hence, they are useful in studying counting problems on elliptic curves.

An old conjecture dating back at least to Hildebrand's work in 1989, asserts that the Möbius function exhibits cancellation on shifted primes; that is, $$\sum_{p\le x}\mu(p+2)=o(\pi(x)),\quad \text{or equivalently}\quad \sum_{n\le x}\Lambda(n)\mu(n+2)=o(x).$$ This is also a well-known model case for the parity problem in sieve theory. In this paper, we prove this conjecture. As an application, we show that each of the two sets $\{p:\, \mu(p+2)=1\}$ and $\{p:\, \mu(p+2)=-1\}$ has asymptotic density $1/2$ among the primes $p$ with $\mu^2(p+2)=1$: \begin{equation*} \lim_{x\to\infty} \frac{\#\{p\le x:\ \mu(p+2)=\kappa\}} {\#\{p\le x:\ \mu^2(p+2)=1\}} =\frac12, \quad \kappa\in\{-1,\,1\}. \end{equation*}

We define genus character $L$-functions of orders in quadratic function fields of odd characteristic. We express each as a product of two quadratic $L$-functions and explicit local factors at primes dividing the conductor, with a possible shift of the variable in one of the two quadratic $L$-functions. This product formula yields meromorphic continuation and functional equations. In the real case, we evaluate special values in terms of continued fractions and derive formulas for the product of class numbers of two imaginary quadratic function fields. These formulas are function field analogues of the Kaneko-Mizuno-Zagier formula.

Shifted convolution sums of $\mathrm{GL}(3)\times\mathrm{GL}(2)$ type have been extensively studied in several important settings, while a number of natural cases remain open. In this paper, we first revisit the shifted convolution sums involving $\lambda_\pi(1,n)$ and the divisor function $d(n)$, improving the recent power saving of $1/42$ obtained by Blomer and Li to $1/22$. We also treat the corresponding sums involving $\lambda_\pi(1,n)$ and $r(n,\mathcal{Q})$, settling a previously open case and obtaining a power saving of $1/22$. Here $\lambda_\pi(1,n)$ denotes the normalized Fourier coefficients of a Hecke–Maass cusp form $\pi$ on $\mathrm{SL}(3,\mathbb{Z})$, and $r(n,\mathcal{Q})$ denotes the number of representations of a positive integer $n$ by a positive definite integral binary quadratic form $\mathcal{Q}$. The proof combines two circle methods with a different organization of the major-arc analysis. A key point is that the nonzero Poisson frequencies are kept together in a single coefficient vector, rather than estimated separately, which allows them to be controlled by a finite matrix norm. The zero Poisson frequency is treated separately: its contribution is independent of the reduced residue class, and its arithmetic mean cancels the constant term in the complementary Jutila weight before absolute values are taken. We make this cancellation explicit, including the relevant normalization and local factors. The same mechanism extends to the binary quadratic-form setting, while the remaining Jutila contribution is handled by the available correlation estimates.

For fixed positive degrees $n,m$, alternating monic reciprocal transformations define a rational discrete dynamical system on the finite-dimensional coefficient space of polynomial pairs. This system arises from congruence jumping in polynomial divisibility problems, where a quotient move changes the polynomial state, but its coefficient dynamics can be studied independently. We prove that every state with nonzero constant terms returns after four alternating updates to its rational gauge classes and classify when this return orbit is realized over $\mathbb Z$. The local obstruction is encoded by endpoint $p$-adic valuation defects. After the first integral block, constant-term valuations evolve by an explicit matrix $M_{n,m}\in\operatorname{SL}_2(\mathbb Z)$ whose trace depends only on $s=nm$. This yields a finite-order, elliptic-type regime for $s<4$, a parabolic dilation regime for $s=4$, and a hyperbolic regime for $s>4$. In the hyperbolic case, invariant thresholds, irrational eigencones, and discrete valley formulas give finite criteria for forward and bi-infinite integral orbits; nevertheless, failure can be delayed arbitrarily long. At the pointwise level, separated auxiliary-relation transport factors through the jumped-variable projection. One jump preserves the visible support rank, while two alternating jumps collapse it to dimension at most one; saturation cannot restore rank. In the quadratic specialization, a rank-two $p$-adic relation lattice gives an exact local cancellation-depth formula and a finite criterion on periodic Vieta orbits. We classify the fixed auxiliary relations that saturate indefinitely on an irreducible six-cycle and construct arbitrarily long persistence on parabolic and fixed hyperbolic orbits.

The Hilbert specialization property of a field, whereby irreducibility of polynomials is preserved under specialization of some of the variables, admits an integral counterpart where specialization takes place inside a ring and irreducibility is required over the ring. It is known that the ring $\mathcal{O}_K$ of integers of any global field $K$ satisfies this integrally Hilbertian property. Using a different approach, we reprove this result and strengthen it by showing that the corresponding integral Hilbert subsets of $\mathcal{O}_K^k$ are not only Zariski-dense, as required by integral hilbertianity, but also of positive density. A central ingredient is a generalization to the ring $\mathcal{O}_K$ of the Ekedahl–Poonen formula originally proved for $\mathbb{Z}$ and $\mathbb{F}_q[t]$. This formula, of independent interest, gives the density of tuples in $\mathcal{O}_K^k$ for which coprime polynomials in $k$ variables and with coefficients in ${\mathcal O}_K$ take coprime values. Our generalization requires estimates for the number of points of bounded height in an arithmetic progression. We obtain such estimates as our third main result, thereby contributing to the theory of heights.

In this paper, we give a family of the arithmetic progressions in which the numbers of odd unimodal and odd strongly unimodal sequences are divisible by 2 and by 4. In particular, we answer a question of Bringmann and Lovejoy on the existence of such arithmetic progressions for odd strongly unimodal sequences modulo 4.

OpenAI (using a non-public "internal frontier model”) has recently (6 October 2026) made a number of remarkable claims, including some claims regarding the quasi-Riemann hypothesis (QRH). While there are good reasons to be cautious regarding these claims, one can nevertheless ask what (if true) their implications would be. While there is widespread agreement that any version of the QRH will provide various improved bounds on the location of the prime numbers, I feel that there is some value in making these improvements explicit. Specifically, if we assume validity of the claimed 7/8 version of the QRH then I shall demonstrate (for all $n\in \mathbb{Z^+}$): 1) the existence of primes between eighth powers, in the interval $(n^8,(n+1)^8)$; 2) the existence of primes in the shorter interval $(n^8,n^8+8n^7)$; 3) the existence of primes in the even shorter interval $(n^8,n^8+n^7).$ Similarly for reals greater than unity $x>1$ one can demonstrate: 4) the existence of primes in the Hoheisel-style interval $(x,x+x^{7/8})$. All of this is contingent on the validity of the claimed 7/8 version of the QRH.

Let $E:y^2=x^3-x$. We prove that $\#E(\mathbf{F}_p)/8$ is squarefree and has at most two prime factors for $\gg X/(\log X)^2$ primes $p\le X$. The proof follows the Gaussian CM sieve framework of Iwaniec and Jiménez Urroz and incorporates the intersection terms identified by Xie. For a squarefree Gaussian ideal $\mathfrak{a}$, we separate its largest rational ideal factor and use the resulting divisibility of both integral coordinates to control the large-factor tail. This transfers an ideal Bombieri–Vinogradov estimate to rational norm divisibility at level $1/2$, with any fixed power of the divisor function as a weight. We prove the corresponding distribution estimate for the switched product sequence and apply the transfer to both sequences in Chen's weighted sieve. The primes counted may be restricted to $p\equiv5\pmod8$; their normalized orders are odd, squarefree, and supported on primes congruent to $1\pmod4$.

For integers $r\ge1$ and $n\ge2$, let $s_r(n)$ be the least $N$ such that every sequence of $N$ vectors in $\Z^r$, each nonzero modulo $n$, contains $n$ terms whose sum is zero modulo $n$ but nonzero modulo $n^2$. Let $w_r(n)$ be the corresponding least $N$ when any nonempty subsequence is allowed. These constants were introduced as zero-sum problems of new types by Z.-W. Sun. We prove $s_r(n)\ge(2^{r+1}-2)(n-1)+1$ by constructing an explicit sequence from a lattice polytope, disproving his conjecture $s_2(n)=4n+1$ for $n\ge4$. We obtain $w_r(2)=2r+1$ by classifying all sequences avoiding such nonempty subsequences modulo $4$. We also prove Sun's conjecture $w_2(n)=4n-3$ for $n=2^k3^\ell p$, where $p$ is a prime and $k,\ell\ge0$ are integers, by classifying the extremal sequences of length $4p-4$ modulo $p^2$ when $n=p$. For every fixed $n\ge2$, we prove $w_r(n)/s(\Z_n^r)=O_n(r^2/2^r)\to0$ as $r\to\infty$, where $s(\Z_n^r)$ is the classical Erdős-Ginzburg-Ziv constant, with an explicit threshold on $r$ for strict inequality. Thus, for every fixed modulus $n\ge2$, Sun's proposed equality fails for sufficiently large $r$.

Erdős, Odlyzko and Sárközy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$. O. Klurman, A. P. Mangerel and J. Teräväinen (2023) have shown that three primes suffice when $q$ is sufficiently smooth. We study the moduli $q=p^\gamma$ that are powers of a fixed prime $p$ and we show that the third prime can be taken subpolynomially small. For any $q_\bullet\le q$ for which $q_\bullet\ge\exp\bigl(C(\log q)^{2/3}(\log\log q)^{4/3}\bigr)$, we give an asymptotic formula for the weighted number of representations of each reduced class as a product of two primes not exceeding $q$ and a third not exceeding~$q_\bullet$. We also show that the conjecture holds for all but a proportion $\exp\bigl(-c(\log q)^{1/3}(\log\log q)^{-1/3}\bigr)$ of the reduced classes. Here $C$ and $c$ are positive constants that depend only on $p$. Both results are unconditional and they rely on a wide zero-free region with no exceptional zeros for the Dirichlet $L$-functions attached to characters modulo $p^\gamma$.We also present conditional versions of these results for arbitrary $q$ under a quasi-Riemann hypothesis.

A Zumkeller number is a positive integer whose set of positive divisors can be partitioned into two disjoint subsets of equal sums. In this paper, we investigate the additive properties of these numbers in conjunction with prime numbers. Specifically, we study the natural density of the exceptional set \(\mathcal{E}\) consisting of all positive integers not expressible as the sum of a prime and a Zumkeller number. Let $\alpha(x) = \#\{n \le x : n \in \mathcal{E}\}$ denote its counting function. We establish a quantitative upper bound for $\alpha(x)$, demonstrating that $\alpha(x) \ll x(\log \log x)^{-A}$, where $A = \log_2\left(\frac{2\pi^2}{2\pi^2-3}\right) \approx 0.2378$, thereby proving that the natural density of this exceptional set is $0$.

For a positive integer $n$, we write $L(n)$ in place of the maximum number of distinct primes whose sum is $n$, assuming there is a way of writing $n$ as a sum of distinct primes, with $L(n) = 0$ otherwise. Again if $n$ admits a partition into distinct primes, then we let $K(n)$ denote the number of parts in the unique partition that has a largest product, with $K(n) = 0$ otherwise. In the latest editions of Guy's classic text on unsolved problems in number theory, the problem as to whether or not $L(n) - K(n)$ is unbounded was left open, and it appears that this has remained open. We solve this problem in the affirmative. Our proof relies, to a large extent, on a consequence of Vinogradov's theorem together with a 1948 result due to Sierpiński on the existence of primes whose gaps to the immediately preceding and following primes both exceed any prescribed positive bound.

In the classical art gallery problem, a guard sees a target when the segment joining them lies entirely within the polygon. We explore what changes when sight may bend along regular arcs on absolutely irreducible curves, but only those defined by primitive integer polynomials of total degree at most \(d\) and coefficient height at most \(H\). We call such curves admissible sight carriers. Arcs can dramatically improve sight. In a one-sided comb, no straight-line guard sees selected targets in two different teeth, yet a single central guard sees a chosen target in every tooth along parabolic carriers. Sight can also fail in two different ways. Geometrically, the polygon may block every sight arc of the allowed degree between two points. For instance, in an alternating comb, the number of guards needed for a row of targets spanning the comb grows linearly with the number of sides at each fixed degree, even for carriers with arbitrary real coefficients. Arithmetically, no admissible carrier may pass through the points at all. Even in the unit square, at fixed \(d\) and \(H\), the proportion of rational targets that must guard themselves tends to one as the height bound grows. An algebraic target may lie on no rational curve of degree at most \(d\), however large \(H\) is. Both failures can be overcome for finite target sets by allowing the degree and coefficient height to grow. We show that every finite set of algebraic targets in a rational polygon becomes visible from any rational interior guard, with explicit bounds. We also construct, for every \(d\), arbitrarily large Galois orbits in the unit square, all of projective height \(4^{d+1}\), whose points must each guard themselves at degree \(d\) for every \(H\). Yet at degree \(d+1\) and coefficient height one, a single guard sees the whole orbit along \(Z(Y-X^{d+1})\).

A set $A$ of integers is an $h$-basis for $n$ if every integer in $[0,n] = \{0,1,2,\ldots,n\}$ belongs to the sumset $hA$, that is, can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. The classical postage stamp problem asks for the largest integer $n_h(k)$ such that $[0,n_h(k)] \subseteq hA$ for some set $A$ of nonnegative integers of size $k+1$. We consider the effect of allowing negative elements. Let $n^{\flat}_h(k)$ be the largest integer such that $[0,n^{\flat}_h(k)] \subseteq hA$ for some set $A$ of integers of size $k+1$ containing at least one negative integer. For $k=2$, we have $n^{\flat}_h(2) < n_h(2)$ if $h$ is odd or $h=4$ and $n^{\flat}_h(2) = n_h(2)$ if $h$ is even and $h \neq 4$. Proofs of these comparison results were autonomously discovered and formally verified in Lean by https://aristotle.harmonic.fun/}{Aristotle}, a formal reasoning agent developed by https://www.harmonic.fun/}{Harmonic}. In this paper we give new proofs of these results.

We prove a folklore conjecture on exponential sums with multiplicative coefficients: if $f$ is multiplicative with $|f(n)|\le1$, $(a,q)=1$, $|\alpha-a/q|\le q^{-2}$ and $q\le N/(\log N)^2$, then \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha)\ll\frac N{\log N}+\frac N{\sqrt q}. \] Complete multiplicativity is not required. We also prove two related conjectured bounds: the refinement in which $q$ is replaced by $q(1+N|\alpha-a/q|)$, when this quantity is at most $N/(\log N)^2$, and the corresponding bound for the sum with logarithmic weights. Finally, we show that the bound can fail for multiplicative $f$ satisfying only $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$, for every fixed $A>1$.

Let $p$ be an odd prime, and let $\rho\colon G\to \GSp_{2n}(\Z_p)$ be an irreducible symplectic residually multiplicity free representation. We define the symplectic extension graph $\Gamma_\rho^\GSp$ associated with $\rho$ to be the directed graph whose vertices are the residual factors of $\rho$ and a directed edge between two vertices corresponds to a nonsplit extension between the vertices realized by $\rho$. Using the Euclidean Bruhat–Tits building associated with $\GSp_{2n}(\Q_p)$, under some assumptions, we prove the existence of certain paths in $\Gamma_\rho^\GSp$ and show that $\Gamma_\rho^\GSp$ is a disjoint union of nonempty strongly connected subgraphs. Moreover, we provide examples to show that the result we obtain is optimal.

The shape of a number field $K$ of degree $n\geq2$ is the similarity class of the lattice obtained by projecting $\mathscr{O}_K$ orthogonally away from the rational line in Minkowski space. It defines a point in the space of shapes of rank-$(n-1)$ lattices, $\mathscr{S}_{n-1}:=\mathrm{GL}_{n-1}(\mathbb{Z})\backslash \mathrm{GL}_{n-1}(\mathbb{R})/\mathrm{GO}_{n-1}(\mathbb{R})$, where $\mathrm{GO}_{n-1}(\mathbb{R})$ is the group of orthogonal similitudes. Let $G$ be a fixed nontrivial finite abelian group, and let $K/\mathbb{Q}$ vary over tamely ramified Galois extensions with Galois group isomorphic to $G$. We study the variation of the shapes of such number fields in $\mathscr S_{n-1}$, where $n=|G|$.

Bachmann and van Ittersum conjectured that every polynomial $f\in\mathbb{Q}[x,y,z]$ satisfying $$ f(x,y,z)=f(y,z,x),\qquad f(x,y,z)+f(-y,x-y,z)+f(y-x,-x,z)=0 $$ is a sum of polynomials in two variables. We prove this conjecture over every field of characteristic zero and determine all homogeneous solutions. Moreover, restriction to $z=0$ identifies the degree $n$ solution space with the space of binary period polynomials of degree $n$ normalized to vanish on both coordinate axes. The solution corresponding to such a polynomial $q$ is $$ f(x,y,z)=q(x,y)+q(y,z)+q(z,x). $$ There are no nonzero homogeneous solutions of odd degree. For even degrees $n$, the symmetric and alternating solution spaces are each isomorphic over $\mathbb{C}$ to the space of weight $n+2$ cusp forms on $SL_2(\mathbb{Z})$. The proof connects the polynomial relations with the cohomology of arithmetic groups through the rational Tits building for $GL_3$. Cusp forms enter both the description of the solutions and the proof that there are no others. The main theorems and the essential argument are formalized unconditionally in Lean~4 by AxiomProver.

In this paper, we study the Chow group of zero-cycles $\text{CH}_0(S)$ for certain Enriques quotients $S$ of the Kummer surface associated to the product of two geometrically non-isogenous elliptic curves $E_1$ and $E_2$. We show that over fields of characteristic different from $2$ or $3$, the degree-zero subgroup $A_0(S) \subset \text{CH}_0(S)$ is 16-torsion. Over p-adic fields, we compute the transcendental Brauer group $\text{Br}(S)/\text{Br}_1(S)$. Using the Brauer-Manin pairing, we construct examples of Enriques surfaces for which $A_0(S)$ attains nontrivial 2-torsion elements arising from push-forward from the abelian surface $E_1\times E_2$.

Let $p$ be a prime $\ge 5$. We determine the semisimplifications of the mod $p$ reductions of irreducible two-dimensional crystalline representations of the Galois group of ${\mathbb Q}_p$ of slope $2$ and weights $k\ge3p+4$ with $k \not \equiv 6 \bmod (p-1)$. The proof relies on the compatibility between the $p$-adic and the mod $p$ Local Langlands Correspondences for $\mathrm{GL}_2(\mathbb{Q}_p)$.