Papers

We prove two conjectures of Samart as identities of genuine Mahler measures. The first is the case $k=1$ of his evaluations for the family $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k^{1/2}$: $m((x+x^{-1})(y+y^{-1})(z+z^{-1})+1)=4L'(g_7,0)$, where $g_7(\tau)=\eta(\tau)^3\eta(7\tau)^3$ is the unique newform of $S_3(\Gamma_0(7),\chi_{-7})$ (LMFDB label 7.3.b.a). The second is the conjugate pair of quadratic CM entries of his 2015 table: $n_2((47\pm 45\sqrt{-7})/2)=(4/7)(54L'(g_7,0)+L'(\chi_{-7},-1))$, where $n_2(s):=2m((x+x^{-1})(y+y^{-1})(z+z^{-1})+\sqrt{s})$. In both cases the parameter lies inside (or on the boundary of) the critical locus, so the passage from the holomorphic (modified) Mahler measure—where Samart computed the $L$-value side conditionally, and Fei the real-part level—to the true Mahler measure was open. For the first theorem we close the gap by a differential-comparison continuation of the Mahler differential along an explicitly certified path, followed by a continuity argument at the interior point. For the second theorem no continuation is needed: the second preimage of the parameter under the modular parametrization is the Fricke partner of Samart's CM point and lies in his proved region, and the evaluation is an exact CM lattice-sum computation. The analytic and arithmetic inputs not proved here are stated explicitly with references; the finite numerical inequalities used in the continuation argument are certified by interval arithmetic, and all identities are confirmed numerically to 41–60 digits.

Let $H_k$ be an $L^2$-normalized Hecke basis for the space of all holomorphic cusp forms of weight $k$. We show that $\max_{f\in H_k}\Vert F\Vert_6\gg (\log\log k)^{\frac{1}{2}}$ where $F(z)=(\Im z)^{\frac{k}{2}}f(z).$ This confirms that the $L^6$-norm of Hecke eigenforms does not converge uniformly as the weight goes to infinity. We also give some results on the joint mass of degree $6$.

These lecture notes are based on three courses given at Nagoya University. Their purpose is to give a beginner-friendly introduction to multiple zeta values and several of their variants, such as finite and symmetric multiple zeta values, q-analogues of multiple zeta values, and multiple Eisenstein series. These notes will be updated in the future.

Let k be a number field. For quotient stacks [X/G] with X a smooth quasi-projective geometrically integral k-variety and G a linear k-group, we extend several relations between local-global obstructions previously known for varieties, showing that étale-Brauer obstruction is the finest among various obstructions (such as iterated descent, finite descent, descent, Brauer-Manin).

For a finite sequence of positive integers $\boldsymbol{a}=(a_1,\dots,a_n)$, the restricted partition function $q_{\boldsymbol{a}}(k)$ denote the number of nonnegative integer solutions to the equation $a_1x_1+a_2x_2+\cdots +a_nx_n=k$. It is proved to be a quasi-polynomial of degree $n-1$. Write $q_{\boldsymbol{a}}(k)=\sum_{j=0}^{n-1}c_j(k)k^j$ with periodic coefficient functions $c_j$, and set $b_m=\#\{i:m\mid a_i\}$. In 2008, Beck, Sam, and Woods conjectured that the minimum period of $c_j(k)$ is $\mathrm{lcm}\{m:b_m>j\}$. In this paper, we derive an exact root-of-unity formula for every coefficient function $c_j(k)$. The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of $c_j(k)$. Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.

No quasiperfect number ($\sigma(n) = 2n + 1$) is known, and its number of distinct prime factors is bounded below; the bound $\omega \ge 7$ of Hagis–Cohen has stood since 1982, obstructed by a family of “deep leaves” on which pure enumeration cannot terminate (the scan bound for the intermediate prime reaches $8 \times 10^8$, and the exponent dimension is unbounded). This paper clears that obstruction with three lemmas at the level of secondary-school algebra — a discriminant criterion, a quadratic-residue sieve, and a multilinear resolver — which eliminate the last prime $q$, the intermediate prime $p$, and the exponent dimension respectively, turning a non-terminating search into a finite decision. On this basis all 381 stems of “$3 \mid n$ and $\omega = 7$” and their $79{,}751{,}212$ deep leaves are eliminated, with the ledger closing exactly and zero solutions throughout; the complementary case “$3 \nmid n$ and $\omega = 7$” collapses to a single stem, which is eliminated directly, so that the proof does not rest on any theorem whose published record we could not independently re-verify. Together with the machine elimination of $\omega \le 6$ (Theorem B4), this yields the main theorem: \emph{any quasiperfect number, if one exists, satisfies $\omega(n) \ge 8$} — the first advance of this bound since Hagis–Cohen 1982. The full computation has been reproduced by seven separately closed ledgers across three algorithmic architectures (CPU and GPU), all with zero solutions and exact ledger closure, and the lemma layer is formalized in Lean (259 theorems, zero sorry). A 2023 preprint of Zemann reported the same bound by a different computation; our audit of its public code found a coverage gap of 35 feasible exponents, so the elimination given here is, to our knowledge, the first complete proof. Code, ledgers, and Lean sources are available from the authors.

The norm map \(N:\mathcal{H}_{\mathbb{Z}_{p^k}}\to \mathbb{Z}_{p^k}\) is studied on the quaternion ring over \(\mathbb{Z}_{p^k}\), where \(p\) is an odd prime and $k\ge 1$ an integer. By means of the isomorphism \(\mathcal{H}_{\mathbb{Z}_{p^k}}\cong M_2(\mathbb{Z}_{p^k})\), quaternions are investigated using matrix methods. It is shown that the fibre size \[ a_{p^k}(m)=|\{q\in \mathcal{H}_{\mathbb{Z}_{p^k}}:N(q)=m\}| \] depends only on the \(p\)-adic valuation \(v_p(m)\) of \(m\). Explicit formulas for the fibre sizes are derived for every \(m\in\mathbb{Z}_{p^k}\): \[ a_{p^k}(m)= \begin{cases} p^{3k-2}(p^2-1), & t=0,\\[6pt] p^{3k-2-t}(p+1)(p^{t+1}-1), & 0<t<k,\\[6pt] p^{2k-1}(p^{k+1}+p^k-1), & t=k, \end{cases} \] where \(t=v_p(m)\) (with the convention \(v_p(0)=k\)). The main result of the paper is a complete solution to the local four-square problem over the ring \(\mathbb{Z}_{p^k}\): the number \(a_{p^k}(m)\) gives the exact number of representations of an arbitrary element \(m\in \mathbb{Z}_{p^k}\) as a sum of four squares, \[ x_1^2+x_2^2+x_3^2+x_4^2=m. \] The proof is purely algebraic; it relies only on matrix theory and Smith normal form, thus avoiding the abstract machinery of number theory. This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections

We study a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family $$ P_m(x_1,\ldots,x_m,u) = \prod_{j=1}^m(1+x_j) + u\prod_{j=1}^m(1-x_j), \qquad m\geq1. $$ Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for $\mathrm m_{\mathbb D}(P_m)$. We prove that, for every fixed $m$, the value $\pi^m\mathrm m_{\mathbb D}(P_m)$ belongs to the algebra of level-$4$ cyclotomic multiple polylogarithm values, and we evaluate the first nontrivial case $m=2$ explicitly in terms of $\operatorname{Li}_4(1/2)$, $\zeta(3)$, Catalan's constant, $\pi$, and $\log2$. We also derive an explicit three-term asymptotic expansion for $\mathrm m_{\mathbb D}(P_m)$ as $m\to\infty$.

For a prime \(p\), let \(A_p=\{k!\pmod p:1\leq k<p\}\). We prove \(|A_p|\gg p^{8/15}\), improving the general lower bound \((\sqrt{2}-o(1))p^{1/2}\). The proof begins with the identity \((n+2)!=(n+1)!+((n+1)!)^2/n!\) in \(\mathbb{F}_p\), which produces many incidences for a family of fractional-linear maps. After Cauchy–Schwarz, the transition maps between two members of this family become affine lines, with multiplicity at most two. The Cartesian-product point-line incidence theorem of Stevens and de Zeeuw then yields the exponent \(8/15\).

We develop a characteristic-uniform arithmetic theory for \[ \mathcal C_d:\quad (u^2+u)(v^2+v)=d. \] For \(d(1-16d)\ne0\), its smooth \((2,2)\)-completion has four rational boundary points forming \(\mathbb Z/4\mathbb Z\), an intrinsic \(D_8\)-action, the inverse \(-(u,v)=(u,-v-1)\), and the native Kummer map \(\kappa_d(u,v)=(u+1)\). Working natively, we derive complete full-point and differential laws, Kummer ladders and recovery, halving, tripling, \(2P+Q\), division polynomials, isogenies, CM endomorphisms, and pairings, with dedicated formulas in characteristics two and three. Classical models provide proof and optimization dictionaries while all endpoints remain native. For Cd25519, the Kummer line is exactly the X25519 line, and a native Segre recoding realizes the optimized complete \(a=-1\) Edwards full-point dependency graph. We further study \[ \begin{gathered} \mathcal C_{a,b,d}:(u^2+u+a)(v^2+v+b)=d,\quad \mathcal T_{a,d}:(u^2+u+a)(v^2+v)=d,\\ \mathcal R_{\tau,\sigma,\kappa}:(x^2-\tau)(y^2-\sigma)=\kappa xy,\quad \mathcal Q_{\alpha,\beta,\gamma}:x^2y^2+\alpha(x^2+y^2)+\beta xy+\gamma=0. \end{gathered} \] For these product, one-sided twisted, reciprocal, and QRT families, we determine their genus-one geometry, finite-field forms, arithmetic, and isogenies. On each smooth QRT fibre, the Vieta–McMillan map is a fixed elliptic translation. Marking \(D\) gives the state \(P\mapsto(\kappa(P),\kappa(P+D))\), with maps realizing \(n\mapsto mn+r\). This yields logarithmic ladders and an elliptic Lucas calculus with nonlinear addition, fast-index doubling, state-division polynomials, and bridges to elliptic divisibility sequences, sigma functions, and elliptic nets. In characteristic two, every ordinary pointed elliptic curve over a perfect field admits the binary state model. We further develop the \(\mathcal C_d\) platform for isogeny-based cryptography.

Let $K$ be an imaginary quadratic field of discriminant~$-d<-11$, and for $\ell\geq1$, let $\psi$ be a Hecke character of $K$ with trivial finite conductor and infinite type~$(2\ell,0)$. In this work we prove that for any prime $p>3$, the quotient $$ \frac{\langle\theta_{\psi},\theta_{\psi}\rangle}{\Omega_{K}^{4\ell}}, $$ is $\lambda$-integral for a prime ideal $\lambda$ over $p$, where $\langle\theta_{\psi},\theta_{\psi}\rangle$ is the Petersson norm of the theta function $\theta_{\psi}=\theta_{\psi}(\tau)$ attached to $\psi$, and $\Omega_{K}$ denotes the Chowla–Selberg period attached to~$K$, and the product $$ \prod_{i=1}^{h_{K}}\frac{\langle\theta_{\psi_{i}},\theta_{\psi_{i}}\rangle}{\Omega_{K}^{4\ell}} $$ is rational, where the product is over all $h_{K}$ of the underlying Hecke characters.

Let $F/\mathbb{Q}_p$ be a finite extension and let $\breve{G}$ be a connected reductive group with connected center satisfying $p>h_{\breve{G}}$ (the Coxeter number). We write down explicit polynomial equations for the reduced Emerton-Gee stacks (the Borel and the twisted Borel versions that jointly cover the entire $\mathcal{X}_{\breve{G},\mathrm{red}}^{\mathrm{EG}}$), by introducing the mod $p$ Weil-Deligne stacks capturing derived structures of $(\varphi, \Gamma)$-modules in reduced families. This allows us to algorithmically compute the set of irreducible components of $\mathcal{X}_{\breve{G},\mathrm{red}}^{\mathrm{EG}}$, thereby establishing its equidimensionality, and the existence of crystalline lifts by showing the total number of irreducible components equals the number of (mod $p$) crystalline (or potentially semistable) components. The last step of the arguments is to interpolate the rigid analytic Weil-Deligne stacks and the mod $p$ Weil-Deligne stacks to deduce the number of potentially semistable components. An initial implementation of the algorithms based on Gröbner basis establishes the existence of crystalline lifts in the $\breve{G}=\mathrm{F}_4$ case for all $F/\mathbb{Q}_p$, and in the remaining $\breve{G}=\mathrm{E}_6, \mathrm{E}_7$, and $\mathrm{E}_8$ cases for all but finitely many $F/\mathbb{Q}_p$.

We prove that every holomorphic differential form of non-top degree on a locally symmetric variety with Baily-Borel boundary of codimension greater than 1 is square-integrable with respect to the invariant metric. This generalizes a theorem of Weissauer in the Siegel modular case.