For any natural number $\ell $ and any prime $p\equiv 1 \pmod{4}$ not dividing $\ell $ there is a Hermitian modular form of arbitrary genus $n$ over $L:=\Q [\sqrt{-\ell}]$ that is congruent to 1 modulo $p$ which is a Hermitian theta series of an $O_L$-lattice of rank $p-1$ admitting a fixed point free automorphism of order $p$. It is shown that also for non-free lattices such theta series are modular forms.
For fixed $q\in\{3,7,11,19, 43,67,163\}$, we consider the density of primes $p$ congruent to $1$ modulo $4$ such that the class group of the number field $\mathbb{Q}(\sqrt{-qp})$ has order divisible by $16$. We show that this density is equal to $1/8$, in line with a more general conjecture of Gerth. Vinogradov's method is the key analytic tool for our work.
For a prime $p$ larger than $7$, the Eisenstein series of weight $p-1$ has some remarkable congruence properties modulo $p$. Those imply, for example, that the $j$-invariants of its zeros (which are known to be real algebraic numbers in the interval $[0,1728]$), are at most quadratic over the field with $p$ elements and are congruent modulo $p$ to the zeros of a certain truncated hypergeometric series. In this paper we introduce "theta modular forms" of weight $k \geq 4$ for the full modular group as the modular forms for which the first dim$(M_k)$ Fourier coefficients are identical to certain…
A primitive root modulo an integer $n$ is the generator of the multiplicative group of integers modulo $n$. Gauss proved that for any prime number $p$ greater than $3$, the sum of its primitive roots is congruent to $1$ modulo $p$ while its product is congruent to $μ(p-1)$ modulo $p$, where $μ$ is the Möbius function. In this paper, we will generalize these two interesting congruences and give the congruences of the sum and the product of integers with the same index modulo $n$.
We prove the non-planarity of a family of 3-regular graphs constructed from the solutions to the Markoff equation $x^2+y^2+z^2=xyz$ modulo prime numbers greater than 7. The proof uses Euler characteristic and an enumeration of the short cycles in these graphs. Non-planarity for large primes would follow assuming a spectral gap, which was the original motivation. For primes congruent to 1 modulo 4, or congruent to 1, 2, or 4 modulo 7, explicit constructions give an alternate proof of non-planarity.
For any $ε>0$, there exists $q_0(ε)$ such for any $q\ge q_0(ε)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three primes, all of which are below $q^{\frac{3}{2}+ε}$. If we restrict our attention to odd moduli $q$ that do not have prime factors congruent to 1 mod 4, we can find such primes below $q^{\frac{11}{8}+ε}$. If we further restrict our set of moduli to prime $q$ that are such that $(q-1,4\cdot7\cdot11\cdot17\cdot23\cdot29)=2$, we can find such primes below $q^{\frac{6}{5}+ε}$. Final…
We confirm, for the primes up to 3000, the conjecture of Bourgain, Gamburd, and Sarnak on strong approximation for the Markoff surface $x^2+y^2+z^2 = 3xyz$ modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that some natural graphs constructed from this equation are asymptotically Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker spectral gap. In both cases, there is close agreement with the Kesten-McKay law for the density of states for random 3-regular graphs. We also study the connectedness of other level sets $x^2+y^2+z^2-3xyz = k$. In the deg…
Let $K = \mathbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A = A(q)$ denote the Gross curve over the Hilbert class field $H$ of $K$. In this note we use Magma to calculate the values $L(E/H, 1)$ for all such $q$'s up to some reasonable ranges for all primes $q$ congruent to $7$ modulo $8$. All these values are non-zero, and using the Birch and Swinnerton-Dyer conjecture, we can calculate hypothetical orders of the Tate-Shafarevich group of $A$ for all such $q$ up to $4831$.
Let F be a non-Archimedean local field of residual characteristic p, and {\ell} be a prime number different from p. We consider the local Jacquet-Langlands correspondence between {\ell}-adic discrete series of GL(n,F) and an inner form GL(m,D). We show that it respects the relationship of congruence modulo {\ell}. More precisely, we show that two integral {\ell}-adic discrete series of GL(m,D) are congruent modulo {\ell} if and only if the same holds for their Jacquet-Langlands transfers to GL(m,D).
We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class, and there exist many such curves splitting the class. We show that this can be false when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class.
We study the Dyson rank function $N(r,t;n)$, the number of partitions with rank congruent to $r$ modulo $t$. We first show that it is monotonic in $n$, and then show that it equidistributed as $n \rightarrow \infty$. Using this result we prove a conjecture of Hou and Jagadeeson on the convexity of $N(r,t;n)$.
We investigate two Stieltjes continued fractions given by the paperfolding sequence and the Rudin-Shapiro sequence. By explicitly describing certain subsequences of the convergents $P_n(x)/Q_n(x)$ modulo $4$, we give the formal power series expansions (modulo $4$) of these two continued fractions and prove that they are congruent modulo $4$ to algebraic series in $\mathbb{Z}[[x]]$. Therefore, the coefficient sequences of the formal power series expansions are $2$-automatic. Write $Q_{n}(x)=\sum_{i\ge 0}a_{n,i}x^{i}$. Then $(Q_{n}(x))_{n\ge 0}$ defines a two-dimensional coefficient sequence $(a…
Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $Γ_0(N)$ which are supported on finitely many square classes modulo $\ell$, in some cases proving that these forms are congruent to the image of a single variable theta series under some number of iterations of the Ramanujan $Θ$-operator. Here, we study the analogous problem for modular forms of half-integral weight on $\operatorname{SL}_{2}(\mathbb{Z})$. Let $η$ be the Dedekind eta function. For a wide range of weights, we prove that every…
Weights of modular forms on $\mathrm{SO}^{+}(2,l)$ and congruences between Eisenstein series and cusp forms of half-integral weight on $\mathrm{SL}_{2}$
Kyoto University（京都大学）；Tokyo Denki University（东京电机大学）；Osaka University（大阪大学）；Hokkaido University（北海道大学）；Muroran Institute of Technology（室兰工业大学）；Tohoku University（东北大学）
Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo 4. We generalize this to an arbitrary (not necessarily commutative) ring of finite rank over the integers using techniques from linear algebra. Our proof, which relies only on elementary matrix identities, is new even in the classical case.
