Papers

We give explicit annular normal forms and classify Frobenius structures for $p$-adic Schwarzian equations defined by developing differentials. For a unit of the Robba ring, reduction of order expresses the associated module as a quadratic rank-one twist of a unipotent extension determined by the residue. Frobenius existence is characterized by a square-class condition. At odd primes, there are exactly four differential-module types, and a uniform dominant-monomial hypothesis gives a Frobenius formula in analytic families across the zero-residue locus. For rational nonintegral annular powers, we obtain a diagonal normal form and a congruence criterion at odd primes; both results hold at every prime when the unit factor is an explicit square. Applied to the reducible modular equations $\theta^2y-\ell^2E_4y/144=0$, with $\gcd(\ell,6)=1$, this gives exactly two annular types at every prime and determines the least $s\geq1$ for which Frobenius under $q\mapsto q^{p^s}$ exists. The subfamily $\ell=12n+1$ has a single annular type. Heine–Stieltjes residue cancellation and nondegenerate Jacobi configurations provide explicit examples. We also prove an obstruction to constant projective Frobenius equivariance of Robba-ring developing maps.

Let $S(Q,M,N,(a_n))=\sum_{q\le Q}\sum_{(a,q)=1}|\sum_{M<n\le M+N}a_ne(an/q^2)|^2$ be Zhao's large sieve sum with square moduli. At the critical point $N=Q^3$ the best known unconditional bound, due to Baier and Zhao (2008), is $S\ll Q^{1/2+\epsilon}N\sum|a_n|^2$, against the conjectured $Q^{\epsilon}N\sum|a_n|^2$, and the exponent $1/2$ has not been lowered since. We prove that, under Hooley's Hypothesis $R^*$ for short Salié sums – square-root cancellation for $\sum_{x_1<n\le x_2}(n/c)e_c(a\bar n+bn)$ over arbitrary subintervals of a period – one has $S\ll Q^{1/2-1/28+\epsilon}N\sum|a_n|^2$ at $N=Q^3$. The key estimate is a bound for the number $P(\alpha)$ of fractions $a/q^2$, $q\le Q$, within $Q^{-3}$ of a point $\alpha$ near $b/r$: we show $P(b/r+z)\ll(Q^{2/3}r^{-1/3}+Q^{1/4})Q^\epsilon$ for every modulus $Q^{1/2+\epsilon}\le r\le Q^{3/2}$, improving Baier's bound $Q^{9/16}r^{-1/8}$ for $r=p,p^2$. The proof rests on one observation: a sum of modular square roots $\sum_{n\in J}e_r(a\sqrt{jn})$ over an interval $J$ is, after completion and an exact evaluation of quadratic Gauss sums at every modulus, $r^{-1/2}$ times a Salié sum of length $r/|J|$, so Hypothesis $R^*$ yields square-root cancellation directly, at every modulus, without Weyl differencing. The relation of $R^*$ to the conjecture of Friedlander and Iwaniec is made precise. The size of the saving is decided by the small moduli $r\le Q^{2/3}$, where the argument is unconditional: for very small moduli we use Cauchy–Schwarz over the residue classes modulo $r$ followed by the second-derivative test, and for $Q^{3/7}\le r\le Q^{2/3}$ an improvement of the complete-sum input to Baier's earlier treatment, proved in the second part of the paper, which also corrects two steps of that treatment. The paper was prepared in collaboration with Claude (Anthropic); Section 1.10 records what each of us contributed.

This work presents theoretical advances in the study of cyclic and quasi-cyclic lattices. First, we provide elementary facts regarding cyclic and quasi-cyclic lattices, and discuss about the cyclicity of some notable lattices. The main contributions of the paper is in the algebraic setting: we investigate cyclic lattices arising from $\mathbb{Z}$-modules in Galois number fields via the Minkowski embedding. We establish necessary and sufficient conditions for an algebraic lattice to be cyclic over both cyclic and general Galois number fields, expressed in terms of naturally associated groups. Moreover, we derive a necessary and sufficient condition for ideal lattices to be cyclic, depending on the factorization of the ideal in the underlying number field.

For a monic irreducible $f \in \mathbb{Z}[x]$ of degree $n$ and an integer $c$, we study the palindromic transform $F(x) = x^n f(x + x^{-1} + c)$, which produces a polynomial of degree $2n$. We give a discriminant formula $\operatorname{disc}(F) = f(c+2)f(c-2)\operatorname{disc}(f)^2$, sufficient conditions for the irreducibility of $F$, and a criterion showing that $F$ is monogenic when $f$ is monogenic and $f(c+2)f(c-2)$ is squarefree, together with a matching non-monogenicity criterion. Iterating the transform yields monogenic number fields of degree $2^k n$ from a fixed base polynomial. As an explicit example, we construct a monogenic field of degree 512 from $x^4 + x + 1$.

In this paper, we study the $p$-adic zeros of the Tribonacci sequence $T_n$, for a prime $p$. In particular, we complete the analysis that was done in a 2024 paper of Yuri Bilu by investigating the case of $p=11$. In addition, we investigate the primes $p$ for which at the first two values $T_n$ and $T_{n-1}$ which are both divisible by $p$, both of these are actually divisible by $p^2$, proving under the assumption of a generalisation of the $abc$ conjecture that there are infinitely many $p$ which are not such.

We prove some residually reducible $p$-adic Galois representations of number fields arise from modular forms. We study the universal deformation ring arising from deformations satisfying the Fontaine-Laffaille condition at primes over $p$. Under certain conditions, we establish the reduced universal deformation ring is a discrete valuation ring. The method uses pseudocharacters and certain bounds on Selmer groups, where the ideal of reducibility as defined by Bellaïche and Chenevier is shown to be maximal and principal. A self-dual assumption is not needed in our argument. The main result on deformations applies to $n$-dimensional representations. In applications, a congruence between Hermitian modular forms due to Klosin is used with our results to prove modularity of some 4-dimensional $p$-adic representations of the imaginary quadratic field $\mathbf{Q}(i)$.

A qKZ-type additive discrete flat connection in characteristic $p$ has $p$-curvature operators. They are commuting automorphisms of the connection. We present a Bethe-ansatz-type construction of eigensections and eigenvalues of the $p$-curvature operators. An eigensection of $p$-curvature operators is a discrete hypergeometric sum over the finite lattice $\mathbb{Z}^r/p\mathbb{Z}^r$. Thus, the $p$-curvature eigensections are constructed by a finite, discrete analogue of a hypergeometric integral.

We study intrinsic Diophantine approximation on the sphere $S^d$, measured by the quality of the best rational approximation available at a given height. We show that, for $d\geq 3$ and along sufficiently sparse sequences of heights, these quantities have a Weibull limit distribution and the associated counting functions are asymptotically Poissonian.

Given a permutation $\tau$ on $n$ letters, we consider lattices spanned by an orbit of one vector $\boldsymbol x$ in $\mathbb R^n$ under the action of $\tau$ by permutation of the coordinates. Such lattices generalize the important class of cyclic lattices and have previously been studied in~\cite{perm}, where a bound on their rank was established. We prove a sufficient condition on $\boldsymbol x$ for this bound to be achieved. We further investigate the structure of such permutation-invariant lattices, proving that they are glued by the permuted vector from the orthogonal cyclic blocks and giving a determinant formula for the lattice in terms of determinants of these blocks and the norm of the permuted vector. In the case $\boldsymbol x$ is an integer vector, these blocks are sublattices of the root lattices $A_k$ in respective dimensions with root lattices themselves and their glued direct sums also realizable by this construction. We also exhibit a glued construction of permutation-invariant algebraic integral lattices from collections of cyclic number fields. Finally, we prove a strengthened version of a previous result of~\cite{lf_ek} on a related construction of well-rounded lattices spanned by sets of algebraic conjugates.

For $r> 2$, we study the moduli space parameterising fixed degree morphisms $\mathbb P^r\to X$ where $X$ is a smooth hypersurface of low degree. More precisely, we prove the following result: let $n\geq 2, e\geq 1$ and $X\subset \mathbb P^{n-1}$ a smooth degree $d\geq 2$ hypersurface over an algebraically closed field of characteristic zero or greater than $d$, then $\mathrm{Mor}_e(\mathbb P^r, X)$ is irreducible of the expected dimension if \[ n> 2^d(d-1)\binom{de+r-1}{r-1}. \] Our result extends the result of Browning-Yamagishi for $r=2$ via multiblock Weyl differencing.

We introduce a new method that brings the combinatorics of hyperplane arrangements into the study of representation zeta functions of compact Lie groups. For the Witten zeta function $\zeta_\Phi(s)$ associated with a root system $\Phi$, our method yields elegant formulas for $\zeta_\Phi(0)$ and $\zeta_\Phi'(0)$ in terms of the exponents of various parabolic subsystems of $\Phi$. Such formulas do not appear to be readily accessible through the conventional analytic techniques in the literature. More generally, the method applies to a broad family of conical zeta functions, expressing these two special values through the Möbius function of the intersection poset of the associated hyperplane arrangement.

Motivated by questions and results of Erdős, Sárközy and Sós (1989), and Khalfalah and Szemerédi (2006), we introduce an ergodic approach to additive problems concerning solutions of equations of the form \[ x+y=\lfloor\alpha(n)\rfloor,\quad x,y\in A,n\in\mathbb{N} \] where $\alpha(t)$ is a sufficiently regular function, such as a polynomial or a logarithmico-exponential function of polynomial growth (e.g. $\alpha(t)=t^{5/2}+t^2\log t$), and $A$ is either a given set of positive upper density, or a cell of a prescribed finite partition of $\mathbb{N}$. We give a complete description of the Hardy field functions of polynomial growth for which these partition and density problems are always solvable. The ergodic approach also applies to the more general problem of finding $x\in A_1,y\in A_2$, where $A_1,A_2$ are two given sets of positive density. This in turn allows us to treat equations of the form $kx+ly=\lfloor\alpha(n)\rfloor$, where $k,l\in\mathbb{Z}$. Some of our results also hold true when $A$ is a subset of the primes with positive relative upper density.

We develop a complex-analytic approach to the model theory of the (real) unrestricted exponential. As a consequence we derive sharp forms of many of the foundational results for the structure ${\mathbb R}^\text{RE}_{\exp}$ (and more general structures). In particular we establish sharp o-minimality, a sharp form of Wilkie's theorem of the complement, a sharp form of Wilkie's conjecture and a sharp form of piecewise definability by terms. Our approach is based on a complexification of the LE-preparation theorem of Lion–Rolin. We also develop a parallel complex theory for ${\mathbb R}_\text{an,exp}$, proving for example that the rational points of height $H$ on a nowhere-dense definable set can be interpolated by an algebraic hypersurface of degree $\text{poly}(\log H)$. This generalizes a theorem of Cluckers–Pila–Wilkie who proved the same statement for ${\mathbb R}_\text{an}^\text{pow}$.

Let $K$ be a Henselian discretely valued field with excellent ring of integers $\mathcal{O}_K$ and algebraically closed residue field $k$. Let $X\to\mathbb{P}^1_K$ be a cyclic cover of degree $n$ prime to the characteristic of $k$. In joint work with Obus and Srinivasan (arXiv:2609.20553), we define an integer called the conductor-discriminant contribution $\text{cdc}(y)$ associated to a multiplicity $2$ point $y$ of the branch divisor of the normalization in $K(X)$ of a regular $\mathcal{O}_K$-model $\mathcal{Y}$ of $\mathbb{P}^1_{K}$; modulo several key results about $\text{cdc}(y)$, we prove a conductor-discriminant inequality for $X$, extending previous work of Ogg, Saito, Liu, Srinivasan, and Obus$-$Srinivasan. In this companion paper, we supply the necessary technical results for $\text{cdc}(y)$. In particular, we show $\text{cdc}(y)$ is non-negative except under highly restrictive conditions on $n$ and the structure of the branch divisor at $y$. Moreover, if $\text{cdc}(y)$ is negative, we show the spectrum of the complete local ring of any point lying over $y$ under the normalization of $\mathcal{Y}$ in $K(X)$ is a rational double point. Along the way, we show the non-negativity of a related quantity, the conductor exponent-discriminant contribution $\text{cedc}(y)$, which is used in our joint work with Obus and Srinivasan to give a new proof of a result of Kohls.

We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers–Ramanujan and Andrews–Gordon identities for every torus-knot singularity $X^a=Y^b$ with coprime $1<a<b.$ For a prime power $q$, let $\mathcal{NC}_n^{a,b}(\mathbb F_q)$ denote the set of pairs of commuting nilpotent $n\times n$ matrices $(A,B)$ over $\mathbb F_q$ satisfying $A^a=B^b$. We establish the threefold equality between their normalized counts, the HJO $q$-series $Z_{a,b}$, and the explicit infinite product $P_{a,b}$: \[ \underbrace{\vphantom{\Bigg|} \prod_{m\geq1}(1-q^{-m}) \Biggl(\sum_{n=0}^{\infty} \frac{\lvert\mathcal{NC}_n^{a,b}(\mathbb F_q)\rvert} {\lvert\operatorname{GL}_n(\mathbb F_q)\rvert}\Biggr) }_{\text{point count}} = \underbrace{\vphantom{\Bigg|}Z_{a,b}(q^{-1}) }_{\text{\(q\)-series}} = \underbrace{\vphantom{\Bigg|}P_{a,b}(q^{-1}) }_{\text{infinite product}}. \] Our main result is a stronger finite identity: the rank $N$ HJO sum equals $(q;q)_N$ times the generating function for balanced cylindric partitions with entries bounded by $N$. Taking $N\to\infty$ yields the HJO conjecture. The proof combines the compositional rational shuffle theorem of Bergeron–Garsia–Leven–Xin and Mellit with a multiplicativity theorem for slope operators and a determinantal model for bounded cylindric partitions, linked by a common $q$-difference equation. The finite identity and the HJO conjecture have been formalized unconditionally in Lean by AxiomProver.

We prove conductor-discriminant inequalities for all $\mathbb{Z}/n$-covers of $\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$, modulo some calculations appearing in work of the third author (arXiv:2609.20585). Specifically, when such a curve $X$ is given by $y^n = f(x)$ with $f(x) \in\mathcal{O}_K[x]$ and $n\mid\text{deg}(f)$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$. This is a direct generalization of previous work of the first two authors on hyperelliptic curves, which in turn generalized work of Ogg, Saito, Liu, and the second author. When $f$ is monic, this strengthens a result of Kohls stating that the conductor exponent of the Jacobian of such a curve is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$.

By introducing new ergodic-theoretic techniques in nilsystems, we determine which infinite sumset configurations occur in $U^k(\Phi)$-uniform and Nil-Bohr sets. To be more precise, our first result associates the degree $k$ of a $U^k(\Phi)$-uniform set with the variety of sumsets it contains, solving a conjecture of Kra, Moreira, Richter and Robertson. Restricting to Nil-Bohr sets we show the existence of infinite sumsets with summands in the shifted primes $\mathbb{P}-1$. As a consequence, we show that for any real polynomial $Q(n)$ with leading irrational coefficient of degree $k$, and any natural numbers $\ell_1, \cdots, \ell_k$ there is an infinite set $P\subset \mathbb{P}$ such that \begin{equation*} Q\Big(\sum_{p \in I} p\Big) \in U \pmod 1 \quad \text{ for all } I \subset P , |I| = \ell_1, \ldots, \ell_k. \end{equation*}

One conjectures that all integers greater than 14 are the sum of an odd prime and two positive squares. We prove that all integers $n$ with $14<n<10^{26}$ can be represented in this manner. Along the way, we investigate sums of a prime and one square and then “bootstrap” to get our results on sums of a prime and two squares. We also show that all integers between $308$ and $10^{16}$ can be represented as the sum of an odd prime and two positive cubes. Our methods require the determination of the integral points on various elliptic curves.

It is well known that the set of Mahler measures of single variable polynomial has limit points of which a list established by D. Boyd and M. Mossinghoff has been extended through approaches based on genetic algorithms. In this paper, we wish to further extend the list of known limit points by adapting a method of missing data restoration.

We determine the order of the low moments of partial sums of random multiplicative functions associated with Euler products of bounded degree whose unitary local parameters satisfy a prime-square cancellation condition, thereby generalizing Harper's seminal work. The result applies to irreducible compact-group representations of unitary or symplectic type, including the higher-rank Sato–Tate model $SU(d)$ and the odd symmetric-power Sato–Tate model $\operatorname{Sym}^{m}SU(2)$.

Let $\mathbb N$ denote the set of nonnegative integers. For an integer $k>1$ and a set $A\subseteq\mathbb N$, let $R_{1,k}(A,n)$ denote the number of solutions of $n=a_1+ka_2$ with $a_1,a_2\in A$. For integers $k>1$ and $t\ge1$, Yang and Chen defined $f_k(t)$ to be the number of sets $A\subseteq\mathbb N$ for which \[ R_{1,k}(A,n)=R_{1,k}(\mathbb N\setminus A,n) \] for all integers $n\ge t$, and asked whether $f_k(t)$ and $f_l(t)$ are eventually equal for any integers $k,l>1$. We prove that \[ f_k(t)\asymp_k \frac{2^t}{t^{k/2}}, \] which gives a negative answer to their problem.

We present a localized rooted-operator argument for Weil positivity in the real odd logarithmic channel, retaining the polar rank-one term throughout. The three gates are structural levels rather than a serial dependency: Gate I is an independent strict cellular/shell positivity theorem and diagnostic; Gate II carries the load-bearing operator argument; Gate III uses compression, closure, and the restricted odd Weil criterion. In Gate II, the inherited parent response is placed in the same post-old-core/common-cut primal form in which the arithmetic mismatch is charged. The audited MASTER-P2 same-block splice combines the aligned reference contribution with the actual Schur response before the target-ground coefficient is used, while the literal-slot/true-ground mismatch is paid explicitly by a leakage term. The source elimination is written as one exact common-source Schur completion on the same harmonic vector, and the endpoint fold is used only as unitary coordinate transport, so it creates no second source debit. An outward-rounded finite computation and analytic tail give a strictly positive MASTER margin for k >= 7. Starting from the rigorously certified physical coefficient-2 endpoint Y=7, Schur induction yields positivity at the integer endpoints; zero-extension gives non-negativity at every finite support radius. Closure and the restricted odd Weil criterion then give the conclusion stated in the main theorem.

We classify, for every prime power $q$ and every $e\geqslant2$, the permutation binomials $X^r(X^{q-1}+a)$ over $\mathbb F_{q^e}$. Writing $\ell_j(q)=(q^j-1)/(q-1)$, such a binomial is a permutation if and only if $\gcd(r,q-1)=1$, $(-a)^{\ell_e(q)}\ne1$, and $r\ell_h(q)\equiv1\pmod{\ell_e(q)}$ for some $1\leqslant h<e$ coprime to $e$. This proves a conjecture of Masuda, Rubio, and Santiago: every permutation binomial of this form arises from $(X^{q^h}+aX)\circ X^r$ for a suitable $h$. We also determine the exact number of distinct permutation functions represented by this family. As a further consequence, we completely classify the broader family $X^r(X^{d(q-1)}+a)$ in the coprime-index case $\gcd(d,\ell_e(q))=1$. The new ingredient in the main classification is the necessity argument: selected Hermite power sums are organized so that Lucas' theorem turns their coefficients into digit conditions; a Farey-guided local argument then forces successive base-$q$ digits, and cyclic rotations yield the inverse congruence. In characteristic $2$, a mod-$4$ lift to an auxiliary ring retains endpoint information lost modulo $2$.

Let $F$ be the unramified quadratic extension of $\mathbb{Q}_p$, and let $P_n(T)$ denote the polynomials arising in the $p$-adic Fourier theory of Schneider and Teitelbaum. We determine explicitly the norms of the functions $P_n(x\Omega)$ on the Banach spaces of locally $F$-analytic functions of fixed order, where $\Omega$ is a Lubin–Tate period. Our computation is based on a recent valuation formula of Ardakov and Berger for the values $P_n(\Omega)$. As an application, we obtain an explicit normalization of the basis constructed by Bannai and Kobayashi, whose elements all have norm one.

In the present paper, we study finite subgroups of the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields from the point of view of mono-anabelian geometry. We shall refer to a finite subgroup of the outer automorphism group of the absolute Galois group of a mixed-characteristic local field as quasi-geometric if its inverse image in the automorphism group of the corresponding absolute Galois group is isomorphic to the absolute Galois group of some mixed-characteristic local field. It is well-known that the image of the natural injective homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the associated absolute Galois group yields a typical example of a quasi-geometric subgroup. The main result of the present paper asserts the existence of non-quasi-geometric finite subgroups of the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields. This result shows that the set of quasi-geometric subgroups of the outer automorphism group of the absolute Galois group of a mixed-characteristic local field cannot be characterized, in general, as the set of all finite subgroups of the outer automorphism group.

Let $D$ be an odd fundamental discriminant, with $D=1$ permitted, and let $r\geq 1$ be fixed. We prove that, for every sufficiently large integer $\ell$ satisfying $(-1)^\ell D>0$, the first $r$ traced diagonal Rankin–Cohen brackets \[ \mathrm{Tr}_1^{|D|}[G_{\ell-2e,D},G_{\ell-2e,D}]_{2e}, \qquad 1\leq e\leq r, \] are linearly independent in $S_{2\ell}(SL_2(\mathbb Z))$. Here $G_{k,D}$ is the Eisenstein series of weight $k$, level $|D|$, and nebentypus $\chi_D$. The Petersson formula of Kayath, Lane, Neifeld, Ni, and Xue then implies that at least $r$ normalized Hecke eigenforms $f\in S_{2\ell}(SL_2(\mathbb Z))$ satisfy $L(f\otimes\chi_D,\ell)\neq 0$. For $D=1$, this gives, for every fixed $r$ and every sufficiently large $K\equiv 0\pmod 4$, at least $r$ level-one eigenforms of weight $K$ with nonzero central value.

For every prime power $q>2$ and every even integer $n\ge4$, we construct an $\mathbb{F}_{q^{2n}}$-maximal curve of genus $(q^2-1)q^n/2$ that is not covered by the Hermitian curve over $\mathbb{F}_{q^{2n}}$. The defining equation also gives a Kummer model for the Beelen–Montanucci curves when $n\ge3$ is odd. We compute the genus for both odd and even $n$ and give a uniform proof of maximality by counting rational places. For $q>2$ and odd $n\ge5$, we also prove that the Beelen–Montanucci curves are not covered by the Hermitian curve, extending the known result for Galois coverings. For every prime power $q$ and $n\ge4$, we also give an explicit automorphism subgroup of order $(q^n+1)q(q^2-1)$.

Let $f_{0,1,4}(n)$ denote the number of partitions of $n$ into parts congruent to 0, 1 or 4 modulo 5 with each part used at most twice, and let $f_{0,2,3}(n)$ be defined analogously for parts congruent to 0, 2 or 3 modulo 5. We prove that for every prime $p \equiv 3 \pmod 4$ there are explicit non-negative integers $a(p)$ and $b(p)$, determined by $20a(p) \equiv -9$ and $20b(p) \equiv -1 \pmod{p^2}$, such that $f_{0,1,4}(p^2m + a(p))$ and $f_{0,2,3}(p^2m + b(p))$ are congruent modulo 3 to $f_{0,1,4}(m)$ and $f_{0,2,3}(m)$ when $p \equiv \pm 1 \pmod 5$, and to $f_{0,2,3}(m)$ and $f_{0,1,4}(m)$ when $p \equiv \pm 2 \pmod 5$. By quadratic reciprocity the two functions are preserved exactly when 5 is a quadratic residue modulo $p$, and interchanged otherwise. The smallest cases are $f_{0,1,4}(9m) \equiv f_{0,2,3}(m)$ and $f_{0,2,3}(9m+4) \equiv f_{0,1,4}(m)$. The proof reduces each generating function modulo 3 to the square of a Rogers-Ramanujan-type theta function by means of $(1-x)^{\ell} \equiv 1-x^{\ell} \pmod{\ell}$ and the Jacobi triple product, and then dissects the resulting binary quadratic form using the fact that $-1$ is a quadratic non-residue modulo $p$. As corollaries we obtain, for each such $p$, a congruence with exceptions on a progression of modulus $p$, and an infinite family of self-similarity congruences on progressions of modulus $p^{dn}$ with $d = 2$ or $d = 4$ and constants $9(p^{dn}-1)/20$ and $(p^{dn}-1)/20$.

In recent work, Hirschhorn and the second author defined $a_k(n)$ to be the number of partitions of $n$ wherein the even parts come in only one color, while the odd parts may be “colored” with one of $k$ colors for fixed $k\geq 1$. This function generalizes the classical partition function and has been of significant interest because it satisfies a number of congruences. Although prior work studying $a_k(n)$ has resulted in congruences in arithmetic progressions in a somewhat ad hoc manner, this work gives a uniform framework for studying congruences modulo 3. This allows us to prove an infinite family of infinite families of non-nested congruences modulo 3.

We prove that the $L$-packets for $p$-adic $\mathrm{G}_2$ constructed by Gan and Savin satisfy the endoscopic character identities. In the course of the proof, we also prove twisted endoscopic character identities for functorial lifts from $\mathrm{G}_2$ to $\mathrm{PGSO}_8$ with triality. As a byproduct, we also prove a global multiplicity formula for the discrete automorphic spectrum of $\mathrm{G}_2$ coming from the cuspidal automorphic spectrum of $\mathrm{PGL}_3$.