Papers

The classical unsolved Gauss circle problem concerns estimating the error between the number of lattice points inside a circle and the area of the circle as its radius tends to infinity. About a century ago, Hardy proposed a conjecture concerning this problem. In this paper, we attempt to provide numerical evidence in support of the Hardy conjecture through large-scale numerical computations.

We give a new Hecke-theoretic method to control the growth of monodromy in $p$-adic families of automorphic Galois representations for unitary groups and quasisplit classical groups.

Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\mathcal{O}_S^\times$ for its unit group. Let $\pi : X \rightarrow \mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\operatorname{Red}(\pi)$ the set of $\alpha \in \mathbb{P}^1(K)$ such that the fibre $\pi^{-1}(\alpha)$ is reducible (i.e. the Galois action on the fibre is intransitive). Hilbert's Irreducibility Theorem asserts that $\operatorname{Red}(\pi)$ is contained in a thin subset of $\mathbb{P}^1(K)$. In this paper we give an explicit description of $\mathcal{O}_S^\times \cap \operatorname{Red}(\pi)$. As an application we prove the following result inspired by a classical theorem of Pólya and Siegel. Let $p_1,\dotsc,p_s$ be rational primes. Let $f \in \mathbb{Q}[x]$.Then the following are equivalent: - There are infinitely many tuples $(e_1,\dotsc,e_s) \in \mathbb{N}^s$ such that the polynomial $f(x)-p_1^{e_1} \cdots p_s^{e_s}$ is reducible. - $f=p_1^{a_1} \cdots p_s^{a_s} g^\ell$ (with $\ell$ prime) or $f=-4 p_1^{a_1} \cdots p_s^{a_s} g^4$ for some $g \in \mathbb{Q}[x]$ and some integers $a_1,\dotsc,a_s$.

Chung, Graham, and Spiro introduced slow Fibonacci walks and used them to partition the integers $n\ge2$ into two sequences, the down-integers and the up-integers. They studied the local spacing of these two sequences and conjectured that their $\ell$-step gap sets agree for every $\ell\ge1$. We show that the conjecture fails at $\ell=4$ by proving \[ 9\in U_4\setminus D_4 . \]

We prove a structure theorem for the dimension towers~$\{d_k(D)\}_{k\geq0}$ which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture over a real quadratic field~$K=\Q(\qD)$. If~$\eps$ denotes the first totally positive power of a fundamental unit of~$K$ and $t_k = \eps^k + \eps^{-k}$ the trace of its $k$-th power, then the SIC dimension tower~$\{d_k=1+t_k\}_{k\geq0}$ is the level~$m=3$ row of an infinite two-dimensional cyclotomic array~$\{\Psi_m(t_k)\}_{m\geq1,k\geq0}$ attached to~$K$, while the auxiliary factors~$(d_k+1)$ and $(d_k-3)$ are its ramified levels $m=2$ and $m=1$. Here~$\Psi_m$ denotes the minimal polynomial of~ $\zeta_m+\zeta_m^{-1} = 2\cos{2\pi/m}$. This construction arose initially from an attempt to formulate relations among SIC dimensions in $q$-algebraic terms. The central object is a single closed composite norm relation for the two-parameter family $c_{m,k}=1-\zeta_m\eps^k$ over the cyclotomic field tower $\{K(\mu_m)\}_{m\geq1}$. This framework sheds new light on the mod-$p$ analogue of Leopoldt's conjecture, by placing the central 3-symmetry of Zauner's conjecture within a broader arithmetic context. Away from the primes dividing~$2mD$, the valuations $v_p(\Psi_m(t_k))$ at every fixed level~$m$ are described exactly in terms of a single local unit valuation, which is then related, through the~$p$-adic class number formula, to the corresponding $p$-adic $L$-value.

We propose a framework for multiple zeta values in $\lambda$-rings that unifies both classical multiple zeta values and multiple $q$-zeta values. As in the classical case, we show that the multiple zeta function defines an algebra homomorphism from the quasi-shuffle algebra to the $\lambda$-ring. We also prove a formula for multiple polylogarithms in $\lambda$-rings, similar to the iterated integral expression in the classical case. As an application, we explicitly describe the algebra of regular multiple $q$-zeta values, compute the corresponding Poincaré series, and exhibit a small spanning set.

In this paper, we address the subconvexity problem in level aspect for symmetric square $L$-functions for cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{Q})$ with a prescribed local ramification at prime $p$. To be more precise, let $\pi$ be a tempered cuspidal automorphic representation of conductor $q(\pi)=p^2$ with a non-quadratic central character of conductor $p$. We prove that if the corresponding local representation $\pi_p$ belongs to a suitable class of representations $\mathcal S$, then \[ L\left(\frac{1}{2},\,\mathrm{Sym}^2\pi\right)\ll_{\varepsilon, \pi_\infty} q(\mathrm{Sym}^2\pi)^{\frac{1}{4}-\frac{1}{168}+o(1)}, \] where implied constant depends polynomially on the spectral parameters of $\pi_\infty$. This is the first instance of level-aspect subconvex bound for $L$-functions of a $\mathrm{GL}_3(\mathbb Q)$ automorphic representation. Our approach is based on the delta-symbol method. Apart from some standard analytic number theoretic tools, Katz's theory of hypergeometric sums, and Deligne's proof of Weil-conjectures play an important role in the proof.

Let $p$ be a prime. We prove nontrivial bounds on short sums of Dirichlet characters mod $p$ evaluated at a class of polynomials, not necessarily homogeneous, in $n$ variables and of degree $k$. For large $n$, we further achieve nontrivial bounds for sums over boxes with side-lengths as short as $p^{1/(k-1)+\varepsilon}$, which breaks past the Burgess barrier of $p^{1/4+\varepsilon}$ as soon as $k\geq 6$. In the proof, we develop a new variation of the Burgess amplification method that reduces the problem to bounding additive character sums. This is the first case in which a Burgess-type method succeeds in the inhomogeneous setting.

We prove that generically on the basic stratum of split unitary Shimura varieties, the universal abelian variety with endomorphism structure and polarization has minimal automorphism group in a precise sense, except for a few degenerate cases. This is a direct analogue of Oort's conjecture on automorphisms of supersingular abelian varieties. On the way, we explicitly compute the generic automorphism group of the universal $p$-divisible group in a given basic isogeny class.

Let $\mu$ be the Möbius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on the minor arcs of the Hardy–Littlewood dissection throughout the range $Q\le N^{1/2}$. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on $q$ and $Q$ but carry a factor $N^{\varepsilon}$. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval $(P,2P]$, where $P$ may be as small as a multiple of $\log N$; a finite Fejér majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.

Let $p$ be an odd prime and let $\mathbf{G}/\mathbb Z$ be a split connected reductive group with $\operatorname{dim} Z(\mathbf{G})\leq1$. Assume that a split maximal torus of $\mathbf{G}$ admits a cocharacter whose pairing with every simple root is odd, and impose an explicit root-theoretic condition on the reduction of $\mathrm{Lie}(\mathbf{G}^{\mathrm{der}})$ modulo $p$. We construct infinitely many continuous representations $\rho:G_{\{p\}}\longrightarrow\mathbf{G}(\mathbb{Z}_p)$ which are unramified at every finite prime different from $p$ and whose images contain a principal congruence subgroup of $\mathbf G^{\mathrm{der}}(\mathbb{Z}_p)$. When the center has dimension one, the representations may be chosen to have open image in $\mathbf{G}(\mathbb{Z}_p)$. The construction continues the author's earlier work on $\mathrm{GL}_n$-valued representations ramified at one prime, but replaces the residual unobstructedness used there by a relative lifting argument. For $\mathbf{G}=\mathrm{GL}_n$ all the required root-theoretic conditions are automatic for every odd prime. We therefore obtain, for every odd $p$ and every $n>1$, infinitely many representations $G_{\{p\}}\to\mathrm{GL}_n(\mathbb{Z}_p)$ with open image. In particular, this removes the weak Vandiver-type hypothesis occurring in the author's previous work, as well as the restriction $p\geq7$ in that construction.

In this paper, we show that $\operatorname{SL}_2(\mathbb{F}_{13})$ and $\operatorname{SL}_2(\mathbb{F}_{19})$ are Galois groups of totally real extensions of $\mathbb{Q}$. For each of these primes $\ell$, we find a polynomial of degree $\ell+1$ whose splitting field has Galois group $\operatorname{PSL}_2(\mathbb{F}_\ell)$. These fields satisfy Böge's criterion and the associated central embedding problem has a proper solution with Galois group $\operatorname{SL}_2(\mathbb{F}_\ell)$. One can choose the resulting fields totally real via a quadratic twist. The degree $14$ polynomial is found through a genus one Hurwitz family of degree 14 covers. The degree 20 polynomial is found using Yang's modular equations for the Shimura curve $X_6^*(1)$.

We consider probability measures on compact subsets of the complex plane arising as limiting distributions of Galois conjugates of certain algebraic integers. These measures are characterized by infinitely many integral inequalities, one for each nonzero integer polynomial. We study a broad family of measures given by particular convex combinations of harmonic and equilibrium measures. We then show that, for measures in this family, it suffices to verify two conditions to guarantee all the required inequalities. Under additional arithmetic constraints on the underlying compact sets, we further show that the infinite collection of inequalities is equivalent to checking the inequalities associated with two explicitly specified integer polynomials.

In this paper, we study the combinatorial object $\bar{b}^k_{r,s}(n)$ which counts the overcolored partition $k$-tuples wherein both even and odd parts are colored with $r$ and $s$ colors, respectively. We extend results of Chacon and Sellers for several families of $r,s$ and $k$. We also establish divisibility properties for $\bar{b}^k_{r,s}(n)$ modulo prime $p$ and conclude with new congruences modulo powers of $2$. The techniques involved to obtain our results rely on theta function identities and modular forms.

The way numbers are represented strongly influences which arithmetic structures are easy to see. The prime clockwork is a recursively growing discrete dynamical system: a list of autonomous two-hand clocks driven by one common $+1$ signal. No primes or primality labels are supplied. Starting empty, the process appends a clock of period $n$ whenever none already present rings; the primes are generated internally as its growth times. For each installed prime $p$, the seconds reading $R_p$ advances through $0,\ldots,p-1$, and each return to zero increments the minutes reading $M_p$, which counts completed $p$-cycles. The hands use only increment, comparison, reset, and carry, without explicit mod or div operations. At time $n$, $n=pM_p(n)+R_p(n)$. The valuation readout $V_p(n)=\nu_p(n)$ is generated locally: it is zero when the seconds counter is non-zero (silent state) and otherwise (when the p-clock rings) one plus the earlier valuation addressed by the current minutes reading. The valuation vector gives the integer in unique prime-factorized form. Its coordinates add and subtract under multiplication and division, representing every positive rational uniquely; divisibility becomes weak componentwise order, and unique factorization is natural in this representation. Finite seconds arrays form Cartesian-product state spaces whose common orbit visits every joint state once before repeating; this grand cycle is the order-sensitive dynamical counterpart of the Chinese remainder theorem. The same coordinates expose gcd, lcm, perfect powers, Bézout's identity, and Euler's totient. Rational valuation levels reach certain positive algebraic irrationalities, but not algebraic numbers in general.

We prove the following inequality for Dirichlet polynomials: \[ \int_0^1 |f(1/2+\sigma)|\,d\sigma\lesssim \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T |f(it)| \, dt. \] In particular, for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ belonging to the Hardy space $\mathscr{H}^1$ of Dirichlet series, \[ \left|a_1+\sum_{n=2}^\infty \frac{a_n}{\sqrt n\log n}\right| \lesssim \|f\|_{\mathscr{H}^1}. \] This answers a question raised previously in the literature and it proves that the multiplicative Hilbert matrix has a bounded symbol.

For a fixed characteristic $p$, let $\delta(p)$ denote the smallest degree needed to guarantee the existence of an isogeny from any supersingular curve $E$ to its Frobenius conjugate $E^{(p)}$. We translate existence questions concerning $\delta(p)$ into questions about positive definite integral ternary quadratic forms and use Voronoi's theory of perfect forms in dimension three to characterize when $\delta(p)$ is nearly maximal. For sufficiently large primes, we show that $\delta(p)$ attains its upper bound precisely when $p$ is represented by one of nine explicit cubic polynomials, reducing the infinitude of such primes to whether one of these cubics takes prime values infinitely often. We also prove that, for almost all primes, $\delta(p)$ lies at least $p^{1/6-o(1)}$ below its upper bound.

We present the discrete–complex complement quotient (DCCQ) framework through the ordered Bernoulli-word kernel $f(p,n,k)=p^k(1-p)^{n-k}$ and its inverse-integer level sets. The binary level $2^{-n}$ selects $p=1/2$ as the unique real split-independent anchor. Imposing the additional continuation rule $1-z=\overline z$ gives the conjugation-symmetric line $z=1/2+iu$. On the central normalized branch, the inverse-level split $\kappa=k/n$ also has real part $1/2$. The quadratic coordinate $Q(z)=1/4+u^2$ identifies complementary conjugate points, has minimum $1/4$, and has an explicit two-sign inverse; restricting its values to integers defines an arithmetic quantization. For supplied positive critical-line zero ordinates $\gamma_k$, the induced level has the exact decomposition \[ L_k=\frac14+\gamma_k^2=N_k+\delta_k,\qquad \delta_k=\widetilde B_1\!\left(\gamma_k^2+\frac34\right), \] where $N_k=\lfloor L_k+1/2\rfloor$ and $\widetilde B_1(x)=\{x\}-1/2$. The integer–phase pair $(N_k,e^{2\pi i\delta_k})$ retains the level and the positive ordinate. Prime-exponent coordinates describe the factorization of the integer component, while conditional Fourier moments formulate an open question about the residual phases. The finite numerical diagnostics are descriptive. A distinct complex exponent $s$ gives an inverse-level representation of the supplied term $m^{-s}$; direct substitution cancels the auxiliary coordinates. These terms assemble into the classical Dirichlet series and Euler product for $\Re s>1$. The framework organizes exact coordinate identities, classical zeta connections, and numerical diagnostics; it establishes neither conditional equidistribution nor a zero-location theorem.

We prove that the algebra of multiple Eisenstein series is an $\mathfrak{sl}_2$-algebra. In particular, we show that it is graded by weight, which also holds after taking complex linear spans. Further, we identify it as a graded $\mathbb{Q}$-algebra with the associated graded algebra of $q$-analogues of multiple zeta values with respect to the weight filtration. The main ingredient is an estimate that compares the usual defining sums of multiple Eisenstein series with sums over a lattice order depending on an integer $N$.

The derived set of multiple zeta-star values is the half-line $[1,+\infty)$. In this paper we study the corresponding two-dimensional problem for multiple star polylogarithms on the unit circle. We first prove that every shifted multiple star polylogarithm is the generating function of a completely monotone sequence and hence admits a normalized Hausdorff–Stieltjes representation. Both the shifted and the unshifted functions, as well as their infinite-depth limits, are shown to be univalent on the half-plane $\mathrm{Re}\,z<1$. For finite indices, the representing densities satisfy a strict monotone likelihood-ratio order with respect to the reverse lexicographic order. Together with closure properties of the Hausdorff–Stieltjes class and a limiting quotient argument, this shows that, along the upper semicircle, the argument of each translated curve increases strictly while its modulus decreases strictly; the corresponding lower-semicircle statement follows by conjugation. The same idea gives strict separation of the curves attached to different indices. We then establish a one-to-one correspondence between a natural set of pairs consisting of a point of the unit circle and an infinite index, and the closed half-plane $\mathrm{Re}\,w\geq \frac12$ with the point $1$ removed. Under a binary coordinate on the index set, this correspondence is a homeomorphism. As an application, a subclass of shifted cyclotomic multiple zeta-star values of all levels form a countable dense subset of the open half-plane $\mathrm{Re}\,w>\frac12$. Consequently, the derived set, and every higher derived set, of this cyclotomic family is the closed half-plane $\mathrm{Re}\,w\geq\frac12$.

We establish a non-Archimedean Caffarelli-Silvestre extension theory for the Vladimirov-Taibleson operator $D^s$ for $s>0$ and, more generally, for hierarchical Laplacians $L_C$. For every $f\in\mathcal{S}(\mathbb{Q}_p^n)$, we equip the Bruhat-tits tree $\mathcal{T}_{p^n}$ with the weight $w_{v_{k-1}v_k}=p^{k(s-n)}$ under the horocyclic coordinates. We prove that the resulting Dirichlet problem admits a unique bounded weighted-harmonic solution that is continuous on the end compactification. Its boundary traces converge to $f$ uniformly and in $L^2$, while the associated deformed normal derivative converges pointwise and in $L^2$ to $D^s f$. We derive explicit Fourier and Poisson representations of the extension and establish an energy identity between its weighted tree energy and the quadratic form of $D^s$. Motivated by $p$-adic AdS/CFT, we also give an equivalent massive formulation on the unweighted tree, based on a renormalized trace and a scale-corrected normal derivative. Finally, under natural local-finiteness and scaling assumptions, we extend the construction to hierarchical Laplacians $L_C$ on ultrametric spaces. We characterize the canonical edge conductances by requiring the cancellation Green operator on the ultrametric tree $\mathcal{T}_X$ to coincide with $L_C^{-1}$ on $\mathcal{S}_0(X)$. In the corresponding canonical flux class, the extension is unique, its Dirichlet-to-Neumann map is $L_C$, and it satisfies the associated energy identity. The Vladimirov-Taibleson construction is recovered as the homogeneous special case.

We study the 2-division of the Klein-Vélu elliptic normal septic. Its three non-zero 2-torsion points give rise to an explicit cubic equation over the function field of $X(7)$. We identify the three roots of this cubic with modular functions expressed in terms of septic theta functions at $\tau/2,\tau$, and $2\tau$, and show that its splitting field is precisely the function field of $X(14)$. This gives an explicit link between the 2-division geometry of the Klein-Vélu septic and the passage from full level 7 to full level $14$. Several intermediate modular function fields in the resulting $S_3$-extension are also described.

A positive $D(n)$-$m$-tuple is a set $A=\{a_1,\ldots,a_m\}$ of distinct positive integers such that $a_i a_j+n$ is a square for every $i\ne j$. In 2005, Dujella and Luca obtained an absolute bound for the cardinality of a $D(p)$- or $D(-p)$-tuple of positive integers, uniformly in the prime $p$. We extend the underlying gap argument to prime powers. An explicit toric elimination certificate, proved by elementary linear algebra, replaces the unsaturated homogeneous elimination step and is valid modulo every prime power. Explicit degree and height bounds for this eliminant yield a uniform gap principle. Combined with the general bound for bounded $|n|$, this shows that every positive reduced $D(\pm p^r)$-tuple (i.e. tuple with elements not divisible by $p$) has less than $2^{121}$ elements, independently of $p$ and $r$. Thus, every positive $D(\pm p)$-tuple has at most $2^{121}$ elements, positive $D(\pm p^2)$-tuples have less than $2^{122}$ elements, and the maximal cardinality of a positive $D(\pm p^r)$-tuple is $O(r)$ uniformly in $p$.

For distinct positive integers v_1, ..., v_n let ML(v_1, ..., v_n) be the largest number L such that at some time t every t vi is at distance at least L from the nearest integer; the Lonely Runner Conjecture asserts that ML >= 1/(n+1). Write ML = p/q in lowest terms. Kravitz conjectured that whenever ML < 1/n one has q = np + 1; Fan and Sun found counterexamples for n = 4, conjectured that q = np + k with 1 <= k <= n always holds, and observed that in their data for n = 6 only k = 1 and k = 3 occur. We explain this observation. For six speeds we show that all but finitely many tuples with ML < 1/6 satisfy ML = (P-1)/(6P) for an integer P congruent to 1 or 5 modulo 6; in particular k is 1 or 3 and the denominator q is odd. The proof determines the three infinite two-parameter families of tuples on which such values concentrate, computes ML exactly on each family, and describes exactly where k = 3 occurs. An exhaustive search over the 2 x 10^9 sextuples with speeds at most 110 finds no exception. For five speeds the same method, together with Chen's classification of the tuples attaining ML = 1/5, shows that all but finitely many tuples with ML < 1/5 satisfy Kravitz's original conjecture. The computations are exact and the code is provided.

Let $\delta\in\mathbb{F}_{2^n}$ satisfy $\operatorname{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(\delta)=1$. We study the permutation behavior of $$ f(x) = \left(\frac{1}{x^2+x+\delta}\right)^{2^k}+x $$ over $\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerical evidence suggests that no other cases occur. In this paper, we confirm their assertion by proving that, for $0\leq k<n$, $f(x)$ is a permutation of $\mathbb{F}_{2^n}$ if and only if $k=0$ or $k=1$.

We establish various connections between classes of colored and bounded ordinary partitions on one hand, and colored but not necessarily bounded binary and $b$-ary partitions on the other hand. Many of these results are based on special recurrence relations, some of which are new. We also obtain several classes of identities for sequences of colored $b$-ary partitions, and there are a few results concerning compositions.

Using BSD results for CM elliptic curves, we relate Qin's quadratic form representation defects to the Cassels pairing on the pure $2$-Selmer group of a congruent number elliptic curve. When the pure $2$-Selmer dimension is even, we prove that the normalized representation defect modulo $2$ is the Pfaffian of the Cassels pairing matrix; the dimension of its radical yields sharper $2$-adic divisibility and information on the $2$-primary Shafarevich–Tate group. For products of primes congruent to $1$ modulo $8$ that are pairwise quadratic residues, we give explicit Cassels pairing matrices for both $E_n$ and $E_{2n}$ and express their entries in terms of quartic and Rédei symbols. For each fixed prime $p$, we construct a governing field of degree $256$ and determine the exact joint distribution of the two Pfaffians by Chebotarev's theorem. In particular, there is a set of primes $q$ of natural density $5/128$ for which both $E_{17q}$ and $E_{34q}$ have rank zero and $2$-primary Shafarevich–Tate group isomorphic to $(\Z/2\Z)^4$.

This exposition constructs group laws by residual intersection on short Weierstrass cubics, the curves $(u^2+u)(v^2+v)=d$, Edwards curves, Hessian curves, Jacobi quartics curves, Jacobi intersections curves, Huff curves, and a symmetric QRT biquadratic. Lines give the operation on plane cubics, while planes give the operation on smooth intersections of two quadrics in $\PP^3$. Associativity is proved by comparing two products of three line or plane equations. After the common zeros are cancelled, equality of the remaining zeros follows from the elementary fact that a smooth projective curve of genus one admits no nonconstant rational function with one simple pole. The formulas for $(u^2+u)(v^2+v)=d$ are valid in every characteristic and are written explicitly in characteristics $2$ and $3$; the other models are treated in characteristic different from $2$ and $3$.

Let $K$ be a field and let $E/K$ be a nonsingular projective curve in general Weierstrass form. We prove associativity of its chord–tangent operation by combining polynomial factorization, formal differentiation, and quadratic interpolation. The main identity eliminates the two nonvertical lines $Y=\ell(X)$ and $Y=m(X)$ occurring in the successive additions $S=P\oplus Q$ and $T=S\oplus R$. There are $c\in K$ and a monic quadratic polynomial $h\in K[X]$ such that \[ \ell+m+\Apol=c(X-x_S), \qquad \Gpol+\ell m=(X-x_S)h, \] and \[ (Y-\ell)(Y-m) =(X-x_S)(h-cY) +Y^2+\Apol(X)Y-\Gpol(X). \] When $c\ne0$, the graph of $q=h/c$ contains $P,Q,R$, and $-T$. The two bracketings determine the same quadratic by ordinary or Hermite interpolation and therefore yield the same final point. The proof applies in every characteristic, since tangency is expressed through the formal partial derivatives of the Weierstrass equation. For the short Weierstrass form, a separate concise proof identifies $q$ with the interpolating parabola in Zwegers's construction.

Let $p$ be a prime and let $\mathcal{P}_p$ be the set of primitive elements of $\mathbb{F}_p$. Inspired by the work of Cohen, Oliveira e Silva and Trudgian on consecutive primitive elements, and by the work of Chang on arithmetic progressions in multiplicative subgroups of finite fields, in this paper, using estimates for multiplicative character sums over systems of linear forms together with the relative Szemerédi's theorem of Conlon, Fox and Zhao, we show that, for every fixed integer $k\ge3$ and any fixed number $0<\delta\le 1$, each subset $A\subseteq\mathcal{P}_p$ with $\#A\ge\delta\#\mathcal{P}_p$ contains a nontrivial $k$-term arithmetic progression, provided that $p$ is sufficiently large. Also, by applying Behrend's theorem, we construct a large subset $A'$ of $\mathcal{P}_p$ with $\#A'=\lfloor(\#\mathcal{P}_p)^{1-\varepsilon}\rfloor$ such that $A'$ contains no nontrivial $k$-term arithmetic progressions.