Related papers: Newforms with rational coefficients
We prove that an additive form of degree $d=2m$, $m$ odd, $m\ge3$, over the unramified quadratic extension $\mathbb{Q}_2(\sqrt{5})$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge 4d+1$. If $3 \nmid d$, then there…
A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be…
In this paper we study general highest weight modules $\mathbb{V}^\lambda$ over a complex finite-dimensional semisimple Lie algebra $\mathfrak{g}$. We present three formulas for the set of weights of a large family of modules…
In this article we present a class of formulas Fn, n in Nat, that need at least 2^n assumptions to be proved in a normal proof in Natural Deduction for purely implicational minimal propositional logic. In purely implicational classical…
We show that, in the space of all totally real fields equipped with the constructible topology, the set of fields that admit a universal quadratic form, or have the Northcott property, is meager. The main tool is a new theorem on the number…
We provide a new construction of rationality for cubic fourfolds via Mori's theory and the minimal model program. As an application, we present the solution of the Kuznetsov's conjecture for $d=42$ (the first open case). Our methods also…
We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by $\pm 1$. We…
We prove the longstanding physics conjecture that there exists a unique two-parameter $\mathcal{W}_{\infty}$-algebra which is freely generated of type $\mathcal{W}(2,3,\dots)$, and generated by the weights $2$ and $3$ fields. Subject to…
We use theory of modular forms to give formulas for $N(1^{l_1},2^{l_2},3^{l_3},6^{l_6};n)$ for all $l_1,l_2,l_3,l_6 \in \mathbb{N}_0$, with $l_1+l_2+l_3+l_6=6$. We also apply our results to write newforms in $S_{3} (\Gamma_0 (24), \chi )$…
Counting the number of Galois orbits of newforms in $S_k(\Gamma_0(N))$ and giving some arithmetic sense to this number is an interesting open problem. The case $N=1$ corresponds to Maeda's conjecture (still an open problem) and the expected…
It is a classical result that there are $12$ (irreducible) rational cubic curves through $8$ generic points in $\mathbb{P}_{\mathbb{C}}^2$, but little is known about the non-generic cases. The space of $8$-point configurations is…
In a recent work we derived the kinematic Hamiltonian and primary constraints of the new general relativity class of teleparallel gravity theories and showed that these theories can be grouped in 9 classes, based on the presence or absence…
Via symbolic computation we deduce 97 new type series for powers of $\pi$ related to Ramanujan-type series. Here are three typical examples: $$\sum_{k=0}^\infty \frac{P(k) \binom{2k}k\binom{3k}k…
Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0) where the \forall-quantifiers run over a total of 8 variables, and where F is a…
Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than…
We study the singular series associated to a cubic form with integer coefficients. If the number of variables is at least $10$, we prove the absolute convergence (and hence positivity) under the assumption of Davenport's Geometric…
We initiate a systematic study of the class of theories without the tree property of the second kind - NTP2. Most importantly, we show: the burden is "sub-multiplicative" in arbitrary theories (in particular, if a theory has TP2 then there…
We study the signs of the Fourier coefficients of a newform. Let $f$ be a normalized newform of weight $k$ for $\Gamma_0(N)$. Let $a_f(n)$ be the $n$th Fourier coefficient of $f$. For any fixed positive integer $m$, we study the…
Inspired by Lehmer's conjecture on the nonvanishing of the Ramanujan $\tau$-function, one may ask whether an odd integer $\alpha$ can be equal to $\tau(n)$ or any coefficient of a newform $f(z)$. Balakrishnan, Craig, Ono, and Tsai used the…
Watkins's conjecture suggests that for an elliptic curve $E/\mathbb{Q}$, the rank of the group $E(\mathbb{Q})$ of rational points is bounded above by $\nu_2 (m_E)$, where $m_E$ is the modular degree associated with $E$. It is known that…