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Fix a prime $p\geq5$, an integer $N\geq1$ relatively prime to $p$, and an irreducible residual global Galois representation $\bar{r}: Gal_{\mathbb{Q}}\rightarrow GL_2(\mathbb{F}_p)$. In this paper, we utilize ghost series to study $p$-adic slopes of $\mathscr{L}$-invariants for $\bar{r}$-newforms. More precisely, under a locally reducible and strongly generic condition for $\bar{r}$: (1) we determine the slopes of $\mathscr{L}$-invariants associated to $\bar{r}$-newforms of weight $k$ and level $\Gamma_0(Np)$, with at most $O(log_pk)$ exceptions; (2) we establish the integrality of these slopes; (3) we prove an equidistribution property for these slopes as the weight $k$ tends to infinity, which confirms the equidistribution conjecture for $\mathscr{L}$-invariants proposed by Bergdall--Pollack recently., Comment: 39 pages; comments are welcome

Large fields (also called ample, anti-mordellic) generalize many fields of classical interest, such as algebraically closed fields, real-closed fields, and $p$-adic fields. In this note we answer a question of Pop by generalizing a result of Fehm and prove that finite unions of affine translates of infinite proper subfields are never diophantine subsets of perfect large fields., Comment: 7 pages, comments welcome!

For a prime p, we study the Galois groups of maximal pro-$p$ extensions of imaginary quadratic fields unramified outside a finite set $S$, where $S$ consists of one or two finite places not lying above $p$. When $p$ is odd, we give explicit presentations of these Galois groups under certain conditions. As an application, we determine the structure of the maximal pro-$3$ extension of $\mathbb{Q}(i)$ unramified outside two specific finite places., Comment: 15 pages,Welcome comments

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n-$th Bernoulli numbers. In this paper, we will generalize this problem by using congruent theory and combinatorial methods, and we get some curious congruences.

In this thesis, we aim to develop p-adic analogs of known results for classical periods, focusing specifically on 1-motives. We establish an integration theory for 1-motives with good reductions, which generalizes the Colmez-Fontaine-Messing p-adic integration for abelian varieties with good reductions. We also compare the integration pairing with other pairings such as those induced by crystalline theory. Additionally, we introduce a formalism for periods and formulate p-adic period conjectures related to p-adic periods arising from this integration pairing. Broadly, our p-adic period conjecture operates at different depths, with each depth revealing distinct relations among the p-adic periods. Notably, the classical period conjecture (Kontsevich-Zanier conjecture over $\bar{\mathbb{Q}}$) for 1-periods fits within our framework, and, according to the classical subgroup theorem of Huber-W\"ustholz for 1-motives, the conjecture for classical periods of 1-motives holds true at depth 1. Finally, we identify three $\mathbb{Q}$-structures arising from $\bar{\mathbb{Q}}$-rational points of the formal p-divisible group associated with a 1-motive $M$ with a good reduction at $p$, and we prove p-adic period conjectures at depths 2 and 1, relative to periods induced by the p-adic integration of $M$ and these $\mathbb{Q}$-structures. Our proof involves a p-adic version of the subgroup theorem that we obtain for 1-motives with good reductions., Comment: This thesis is submitted by the author in 2024 in partial fulfillment of the requirements for the degree of Ph.D. in Mathematics at the University of Ottawa

In 2014, as part of a larger study of overpartitions with restrictions of the overlined parts based on residue classes, Munagi and Sellers defined $d_2(n)$ as the number of overpartitions of weight $n$ wherein only even parts can be overlined. As part of that work, they used a generating function approach to prove a parity characterization for $d_2(n)$. In this note, we give a combinatorial proof of their result and extend it to a modulus 4 characterization; we provide both generating function and combinatorial proofs of this stronger result. The combinatorial arguments incorporate classical involutions of Franklin, Glaisher, and Sylvester, along with a recent involution of van Leeuwen and methods new with this work., Comment: 15 pages, 8 tables, 2 figures

Let $\displaystyle \sum_{i=a}^b \frac{1}{i} = \frac{u_{a,b}}{v_{a,b}}$ with $u_{a,b}$ and $v_{a,b}$ coprime. Erd\H{o}s and Graham asked the following: Does there, for every fixed $a$, exist a $b$ such that $v_{a,b} < v_{a,b-1}$? If so, what is the least such $b = b(a)$? In this paper we will investigate these problems in a more general setting, answer the first question in the affirmative and obtain the bounds $a + 0.54\log(a) < b(a) \le 4.374(a-1)$, which hold for all large enough $a$., Comment: 57 pages

We determine all possible degrees of cyclic isogenies of non-CM elliptic curves with rational $j$-invariant over number fields of degree $p$, where $p$ is an odd prime. The question had been answered for $p=2$, so this paper completes the classification in case when the degree of the number field is prime.

Let $E/F$ be a quadratic extension of totally real number fields. We show that the generalized Hirzebruch-Zagier cycles arising from the associated Hilbert modular varieties can be put in $p$-adic families. As an application, using the theory of base change, we give a geometric construction of the multivariable $p$-adic adjoint $L$-function twisted by the Hecke character of $E/F$, attached to Hida families of Hilbert modular forms over $F$., Comment: 55 pages, comments welcome

A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of $b^\prime_5(n)$. We provide full characterization of the parity of $b^\prime_5(2n+1)$, present several congruences modulo 4, and prove that the generating function of the sequence $(b^\prime_5(5n+1))$ is lacunary modulo any arbitrary positive powers of 5., Comment: To appear in the International Journal of Number Theory

We study local-global principles for semi-integral points on orbifold pairs of Markoff type. In particular, we analyse when these orbifold pairs satisfy weak weak approximation, weak approximation and strong approximation off a finite set of places. We show that Markoff orbifold pairs satisfy the semi-integral Hasse principle and we measure how often such orbifold pairs have strict semi-integral points but the corresponding Markoff surface lacks integral points., Comment: 28 pages

We construct several $C^*$-algebras and spectral triples associated to the Berkovich projective line $\mathbb{P}^1_{\mathrm{Berk}}({\mathbb{C}_p})$. In the commutative setting, we construct a spectral triple as a direct limit over finite $\mathbb{R}$-trees. More general $C^*$-algebras generated by partial isometries are also presented. We use their representations to associate a Perron-Frobenius operator and a family of projection valued measures. Finally, we show that invariant measures, such as the Patterson-Sullivan measure, can be obtained as KMS-states of the crossed product algebra with a Schottky subgroup of $\mathrm{PGL}_2(\mathbb{C}_p)$.

Let $F$ be a multi-quadratic totally real number field. Let $\sigma_1,\dots, \sigma_r$ denote its distinct embeddings. Given $s \in F,$ we give an explicit formula for $\| \sigma(s)\|$ and $\sum_{i

We study integrals appearing in one-loop amplitudes in string theory, and in particular their analytic continuation based on a string theoretic analog of the $i\varepsilon$-prescription of quantum field theory. For various zero- and two-point one-loop amplitudes of both open and closed strings, we prove that this analytic continuation is equivalent to a regularization using generalized exponential integrals. Our approach provides exact expressions in terms of the degeneracies at each mass level. For one-loop amplitudes with boundaries, our result takes the form of a linear combination of three partition functions at different temperatures depending on a variable $T_0$, yet their sum is independent of this variable. The imaginary part of the amplitudes can be read off in closed form, while the real part is amenable to numerical evaluation. While the expressions are rather different, we demonstrate agreement of our approach with the contour put forward by Eberhardt-Mizera (2023) following the Hardy-Ramanujan-Rademacher circle method., Comment: 33 pages + appendices

This paper describes the range of cardinalities of sumsets and restricted sumsets of finite subsets of the integers and of infinite abelian groups that are not of bounded exponent., Comment: 7 pages; revised and corrected

The object of observation in present paper is statistical independence of real sequences and its description as independence with re spect to certain class of densities.

The list of all groups that can appear as torsion subgroups of elliptic curves over number fields of degree $d$, $d=4,5,6$, is not completely determined. However, the list of groups $\Phi^{\infty}(d)$, $d=4,5,6$, that can be realized as torsion subgroups for infinitely many non-isomorphic elliptic curves over these fields are known. We address the question of which torsion subgroups can arise over a given number field of degree $d$. In fact, given $G\in\Phi^{\infty}(d)$ and a number field $K$ of degree $d$, we give explicit criteria telling whether $G$ is realized finitely or infinitely often over $K$. We also give results on the field with the smallest absolute value of its discriminant such that there exists an elliptic curve with torsion $G$. Finally, we give examples of number fields $K$ of degree $d$, $d=4,5,6$, over which the Mordell-Weil rank of elliptic curves with prescribed torsion is bounded from above.

We obtain formulae for the numbers of isomorphism and conjugacy classes of non-identity proper subgroups of the groups $G={\rm PSL}_2(p)$, $p$ prime, and for the numbers of those conjugacy classes which do or do not consist of self-normalising subgroups. The formulae are used to prove lower bounds $17$, $18$, $6$ and $12$ respectively satisfied by these invariants for all $p>37$. A computer search carried out for a different problem shows that these bounds are attained for over a million primes $p$; we show that if the Bateman--Horn Conjecture is true, they are attained for infinitely many primes. Also, assuming no unproved conjectures, we use a result of Heath-Brown to obtain upper bounds for these invariants, valid for an infinite set of primes $p$., Comment: 10 pages

We compute the stable wave front set of theta representations for certain tame Brylinski-Deligne covers of a connected reductive $p$-adic group. The computation involves two main inputs. First we use a theorem of Okada, adapted to covering groups, to reduce the computation of the wave front set to computing the Kawanaka wave front set of certain representations of finite groups of Lie type. Second, to compute the Kawanaka wave front sets we use Lusztig's formula. This requires a careful analysis of the action of the pro-$p$ Iwahori-Hecke algebra on the theta representation, using the structural results about Hecke algebras developed by Gao-Gurevich-Karasiewicz and Wang., Comment: Main text is 31 pages, followed by 12 pages of tables. 10 tables

Erdos conjectured that every odd number greater than one can be expressed as the sum of a squarefree number and a power of two. Subsequently, Odlyzko and McCranie provided numerical verification of this conjecture up to $10^7$ and $1.4\cdot 10^9$. In this paper, we extend the verification to all odd integers up to $2^{50}$, thereby improving the previous bound by a factor of more than $8\cdot 10^5$. Our approach employs a highly parallelized algorithm implemented on a GPU, which significantly accelerates the process. We provide details of the algorithm and present novel heuristic computations and numerical findings, including the smallest odd numbers $<2^{50}$ that require a higher power of two as all smaller ones in their representation.

In this note, we generalise two results on prime numbers in short intervals. The first result is Ingham's theorem which connects the zero-density estimates with short intervals where the prime number theorem holds, and the second result is due to Heath-Brown and Iwaniec, which derives the weighted zero-density estimates used for obtaining the lower bound for the number of primes in short intervals. The generalised versions of these results make the connections between the zero-free regions, zero-density estimates, and the primes in short intervals more transparent. As an example, the generalisation of Ingham's theorem implies that, under the Density Hypothesis, the prime number theorem holds in $[x - \sqrt{x}\exp(\log^{2/3+\varepsilon}x), x]$, which refines upon the classic interval $[x - x^{1/2+ \varepsilon}, x]$., Comment: 15 pages

This paper uncovers a new link between the p-adic weight of the support \Supp(f) of f(x)=\sum_{i\ge 0} a_i x^i and the first slope of Artin-Schreier curve X_f: y^p-y=f. We prove that the family X_f has its first slope \ge 1/\max_{i\in\Supp(f)}s_p(i), where s_p(i) is the sum of p-adic digits of i. If the maximum is achieved at a unique i, we show that the lower bound is achieved if and only if \nu satisfies a combinatorial $p$-symmetric condition. As an application, we construct explicit families of curves in every characteristic p with first slope equal to 1/n for every n\ge 2., Comment: 21 pages

We study $\ell$-isogeny graphs of ordinary elliptic curves defined over $\mathbb{F}_q$ with an added level structure. Given an integer $N$ coprime to $p$ and $\ell,$ we look at the graphs obtained by adding $\Gamma_0(N),$ $\Gamma_1(N),$ and $\Gamma(N)$-level structures to volcanoes. Given an order $\mathcal{O}$ in an imaginary quadratic field $K,$ we look at the action of generalised ideal class groups of $\mathcal{O}$ on the set of elliptic curves whose endomorphism rings are $\mathcal{O}$ along with a given level structure. We show how the structure of the craters of these graphs is determined by the choice of parameters.

We define two $L$-functions associated to a common vector valued eigenform $f$ transforming with the ``finite'' Weil representation. The first one can be seen as a standard zeta function defined by the eigenvalues of $f$. The second one can be interpreted as standard $L$-function defined as an Euler product where each $p$-factor is a rational function in terms of two unramified characters of the $p$-adic field $\Q_p$. We show that both $L$-functions are related and prove further that they both can be continued meromorphically to the whole complex $s$-plane., Comment: This is the second part of a series of two articles

In this paper, we contribute to previously known results on lattices constructed by algebraic function fields, or function field lattices in short. First, motivated by the non-well-roundedness property of certain hyperelliptic function field lattices (Ates and Stichtenoth, 2016), we explore the successive minima of these lattices in detail. We also study the determinant of hyperelliptic function field lattices. Finally, we show a connection between the automorphism groups of algebraic function fields and function field lattices, based on ideas from B\"ottcher et al., 2016.

Gcd-graphs over the ring of integers modulo $n$ are a simple and elegant class of integral graphs. The study of these graphs connects multiple areas of mathematics, including graph theory, number theory, and ring theory. In a recent work, inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We discover that, in both cases, gcd-graphs share many similar and analogous properties. In this article, we extend this line of research further. Among other topics, we explore an analog of a conjecture of So and a weaker version of Sander-Sander, concerning the conditions under which two gcd-graphs are isomorphic or isospectral. We also provide several constructions showing that, unlike the case over $\mathbb{Z}$, it is not uncommon for two gcd-graphs over polynomial rings to be isomorphic., Comment: Comments are welcome!

Primality testing is an especially useful topic for public-key cryptography. In this paper, a novel primality test algorithm based on the Pell's cubic will be introduced, and its necessary primality conditions will be proved using three integer sequences connected to operations applied in the projectivization of the Pell's cubic. The number of operations involved in the test grows linearly with respect to the bit-length $\log_2(n)$ of the input integer $n$. The algorithm is deterministic for integers less than $2^{32}$., Comment: 15 pages, see https://github.com/Luckydd99/Novel_Performant_Primality_Test.git for the Python implementation of the primality test

The Zero-free regions of the Riemann zeta function play a pivotal role in understanding the distribution of prime numbers. These regions are often established using specific trigonometric polynomials and associated functionals. In this paper, we revisit the exact values of the infimum $V_n$ of a particular functional $v(f)$ over the set $C_n$ of cosine polynomials with nonnegative coefficients satisfying certain conditions, for $n = 2, 3, 4, 5, 6$. By refining previous methods and classifying the polynomials based on the location of their roots, we provide improved calculations of $V_n$ for these values of $n$. Additionally, we extend our techniques to determine the exact values of $V_n$ for $n = 7, 8$, which were previously unknown. Our findings yield tighter bounds for the constant $R$ in the classical zero-free region $\sigma > 1 - \frac{1}{R \log |t|}$, thereby enhancing our comprehension of the zeros of the Riemann zeta function and contributing to advancements in analytic number theory and prime number theory., Comment: 71 pages, 23 figures

Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for $k \geq 4$ distinct subsets of primes. This extends the work of H.~Li, H.~Pan, as well as X.~Shao on sums of three primes, and A.~Alsteri and X.~Shao on sums of two primes. The primary new contributions come from elementary combinatorial lemmas., Comment: 17 pages

The Subspace Theorem due to Schmidt (1972) is a broad generalisation of Roth's Theorem in Diophantine Approximation (1955) which, in the same way as the latter, suffers a notorious lack of effectivity. This problem is tackled from a probabilistic standpoint by determining the proportion of algebraic linear forms of bounded heights and degrees for which there exists a solution to the Subspace Inequality lying in a subspace of large height. The estimates are established for a class of height functions emerging from an analytic parametrisation of the projective space. They are pertinent in the regime where the heights of the algebraic quantities are larger than those of the rational solutions to the inequality under consideration, and are valid for approximation functions more general than the power functions intervening in the original Subspace Theorem. These estimates are further refined in the case of Roth's Theorem so as to yield a Khintchin-type density version of the so-called Waldschmidt conjecture (which is known to fail pointwise). This answers a question raised by Beresnevich, Bernik and Dodson (2009).

We derive formulas for the number of points on the basic stratum of certain Kottwitz varieties in terms of automorphic representations and certain explicit polynomials, for which we present efficient algorithms for computation. We obtain our results using the trace formula, base change, representations of general linear groups over p-adic fields, and a truncation of the formula of Kottwitz for the number of points on Shimura varieties over finite fields., Comment: arXiv admin note: text overlap with arXiv:1111.6830 by other authors

In this article we derive, using the Lagrange inversion theorem and applying twice the Fa\`a di Bruno formula, an expression of the minimum of the Gamma function $\Gamma$ as an expansion in powers of the Euler-Mascheroni constant $\gamma$. The result can be expressed in terms of values the Riemann zeta function $\zeta$ of integer arguments, since the multiple derivative of the digamma function $\psi$ evaluated in $1$ is precisely proportional to the zeta function. The first terms (up to $\gamma^6$) were provided in order to address the convergence of the series. Applying the Lagrange inversion theorem at the value $3/2$ yields more accurate results, although less elegant formulas, in particular because the digamma function evaluated in $3/2$ does not simplify.

We demonstrate the existence of $K$-multimagic squares of order $N$ consisting of distinct integers whenever $N>2 K(K+1)$. This improves upon our earlier result in which we only required $N+1$ distinct integers. Additionally, we present a direct method by which our analysis of the magic square system may be used to show the existence of magic squares consisting of distinct $k$ th powers when $$ N> \begin{cases}2^{k+1} & \text { if } 2 \leqslant k \leqslant 4, \\ 2\lceil k(\log k+4.20032)\rceil & \text { if } k \geqslant 5,\end{cases} $$ improving on a recent result by Rome and Yamagishi.

We consider the generating series of special cycles on $\mathcal{A}_1(N)\times \mathcal{A}_g(N)$, with full level $N$ structure, valued in the cohomology of degree $2g$. The modularity theorem of Kudla-Millson for locally symmetric spaces implies that these series are modular. When $N=1$, the images of these loci in $\mathcal{A}_g$ are the $d$-elliptic Noether-Lefschetz loci, which are conjectured to be modular. In the appendix, it is shown that the resulting modular forms are nonzero for $g=2$ when $N\geq 11$ and $N\neq 12$., Comment: 25 pages including appendix, comments welcome

Following Zagier, this work studies the rationality and divisibility of Fourier coefficients of meromorphic Hilbert modular forms associated with real quadratic fields, using theta lifts and weak Maass forms. We establish conditions where these coefficients are rational with bounded denominators and demonstrate divisibility properties under suitable linear combinations.

In the following we consider Diophantine equations of the form $x^2+ zxy + y^2 = M$ for given $M,z \in \mathbb{Z}$ and discuss the number of its (primitive) solutions as well as the construction of them. To reach this goal we introduce $z$-rings which turn out to be a useful tool to investigate these Diophantine equations. Moreover, we will extend these rings and study the algebraic curves defined by them on a plane by methods inspired by the complex plane. Then we define the so called subbranches which are bounded and connected parts of the algebraic curves containing a representative of each solution of the Diophantine equations with respect to association in $z$-rings. With the help of them we can easily prove the existence or non-existence of solutions to the above Diophantine equations. Then we divide the integer primes with respect to the different $z$-rings into two main categories, i.e. the regular and irregular elements. We show that the irregular elements are prime in the corresponding $z$-rings and we identify that most of the $z$-rings cannot be unique factorization domains. We determine the number of positive, primitive solutions of the above Diophantine equation if $M \in \mathbb{N}$ is a product of irregular elements in the corresponding $z$-ring for $z \in \mathbb{N}$. We also give an overview how many primitive and non-primitive solutions in a given quadrant we can find for arbitrary $M,z \in \mathbb{Z}$, especially, if $M$ is a power of any irregular element. Furthermore, we consider the case $z = 3$, determine the regular and irregular elements as well as the number of positive, primitive solutions of the Diophantine equation $x^2 + 3xy + y^2 = M$ depending on $M \in \mathbb{N}$., Comment: 68 pages, 13 figures

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for ${\mathbb Z}_K$, the ring of integers of $K={\mathbb Q}(\theta)$, where $f(\theta)=0$. For $n\ge 2$, we define the reciprocal quintinomial \[{\mathcal F}_{n,A,B}(x):=x^{2^n}+Ax^{3\cdot 2^{n-2}}+Bx^{2^{n-1}}+Ax^{2^{n-2}}+1\in {\mathbb Z}[x].\] In this article, we extend our previous work on the monogenicity of ${\mathcal F}_{n,A,B}(x)$ to treat the specific previously-unaddressed situation of $A\equiv B\equiv 1\pmod{4}$. Moreover, we determine the Galois group over ${\mathbb Q}$ of ${\mathcal F}_{n,A,B}(x)$ in special cases., Comment: arXiv admin note: text overlap with arXiv:2404.17921

We prove new bounds for the Fourier coefficients of Jacobi forms using a method of Iwaniec. In view of the Fourier-Jacobi expansion of degree two Siegel modular forms, we can use these to obtain strong bounds on fundamental Fourier coefficients of Siegel modular forms., Comment: 23 pages

In order to study the analytic properties of the Goldbach generating function we consider a smooth version, similar to the Chebyshev function for the Prime Number Theorem. In this paper we obtain explicit numerical estimates for the average order of its summatory function both in the classical case and in arithmetic progressions. These support the existing asymptotic results, under the (Generalised) Riemann Hypothesis, involving error terms.

A graph is called integral if its eigenvalues are integers. In this article, we provide the necessary and sufficient conditions for a Cayley graph over a finite symmetric algebra $R$ to be integral. This generalizes the work of So who studies the case where $R$ is the ring of integers modulo $n.$ We also explain some number-theoretic constructions of finite symmetric algebras arising from global fields, which we hope could pave the way for future studies on Paley graphs associated with a finite Hecke character., Comment: Comments are welcome!

In this note we prove a conjecture by Constantin--Strauss--V\u{a}rv\u{a}ruc\u{a} related to the finite depth water wave problem, tightening their results. The proof uses identities related to Jacobi Theta functions. We also discuss potential implications of the improvement., Comment: 5 pages, 1 figure

We employ Weyl's method and Vinogradov's method to analyze skew-shift dynamics on semi-algebraic sets. Consequently, we improve the quantum dynamical upper bounds of Jitomirskaya-Powell, Liu, and Shamis-Sodin for long-range operators with skew-shift potentials., Comment: 23 pages, comments welcome!

We extend the results of [CGLS22] to higher weight modular forms and prove a rank $0$ Tamagawa number formula (also known as the Bloch-Kato conjecture) for modular forms at good Eisenstein primes. Under standard hypotheses (i.e. the injectivity of the $p$-adic Abel-Jabobi map and the non-degeneracy of the Gillet-Soul\'e height pairing), we also discuss some partial results towards a rank $1$ result. A conditional higher weight $p$-converse theorem to Gross-Zagier-Zhang-Kolyvagin-Nekov\'a\v{r} is also obtained as a consequence of the anticyclotomic Iwasawa Main Conjectures., Comment: 32 pages. Comments welcome!

We study the index of nilpotency relative to certain Hecke operators in spaces of modular forms with integer weight and level $N$ with integer coefficients modulo primes $p$ for $(p, N) \in \{(3, 1), (5, 1), (7, 1), (4, 3)\}$. In these settings, we prove upper bounds on certain indices of nilpotency. Our numerical computations suggest precise formulas analogous to the formula of Nicolas and Serre when $p = 2$ and $N = 1$. As applications, we prove infinite families of congruences for $p^t$-core partition functions modulo $p$ for $p\in \{3, 5, 7\}$ and $t\geq 1$, and we prove an infinite family of congruences modulo $3$ for the $r$th power partition function, $p_r(n)$, when $r = 12k$ with $\gcd(k,6) = 1$., Comment: 19 pages, comments are welcome

Results about the structure of the set of Egyptian fractions on the line are extended to subsets of topological groups.

We prove a Galois equidistribution result for torsion points in $\mathbb G_m^n$ in the $p$-adic setting for test functions of the form $\log |F|_p$ where $F$ is a nonzero polynomial with coefficients in the $p$-adic numbers. Our result includes a power saving quantitative estimate of the decay rate rate of the equidistribution. As an application we show that Ih's Conjecture is true for a class of divisors of $\mathbb G_m^n$., Comment: Comments are welcome

Let $G$ be a finite abelian $p$-group. We count $G$-extensions of global rational function fields $\mathbb F_q(T)$ of characteristic $p$ by the degree of what we call their Artin-Schreier conductor. The corresponding (ordinary) generating function turns out to be rational. This gives an exact answer to the counting problem, and seems to beg for a geometric interpretation. This is in contrast with the generating functions for the ordinary conductor (from class field theory) and the discriminant, which in general have no meromorphic continuation to the entire complex plane., Comment: 11 pages

In this paper we present families of wild 1-motives, i.e., families of pairwise non-isomorphic Deligne 1-motives, over rings of $S$-integers $\mathcal{O}_{F,S}$, which have the same reductions to torsion 1-motives for all $v\notin S$. Our proof is based on a technical result concerning a local to global principle for multiple base discrete logarithm problem for arbitrary big bases., Comment: 12 pages

In an infinitesimal variant of Guo-Jacquet trace formulae, the regular semi-simple terms are expressed as noninvariant weighted orbital integrals on two global infinitesimal symmetric spaces. We prove some relations between the Fourier transforms of invariant weighted orbital integrals on the corresponding local infinitesimal symmetric spaces. These relations should be useful in the noninvariant comparison of the infinitesimal variant of Guo-Jacquet trace formulae., Comment: 42 pages

Let $K$ be an absolutely unramified $p$-adic field. We establish a ramification bound, depending only on the given prime $p$ and an integer $i$, for mod $p$ Galois representations associated with Wach modules of height at most $i$. Using an instance of $q$-crystalline cohomology (in its prismatic form), we thus obtain improved bounds on the ramification of $\mathrm{H}^{i}_{et}(X_{\mathbb{C}_K}, \mathbb{Z}/p\mathbb{Z})$ for a smooth proper $p$-adic formal scheme $X$ over $\mathcal{O}_K$, for arbitrarily large degree $i$., Comment: 14 pages; comments welcome