# Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

2,567 questions
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(1) In "Esquisse d'un programme", Grothendieck conjectures Grothendieck-Teichmuller conjecture: the morphism $$G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})$$ is an isomorphism. Here $G_{\... 0answers 2k views ### The exponent of Ш of$y^2 = x^3 + px$, where$p$is a Fermat prime For$d$a non-zero integer, let$E_d$be the elliptic curve $$E_d : y^2 = x^3+dx.$$ When we let$d$be$p = 2^{2^k}+1$, for$k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD, $$\# Ш(E_p)... 0answers 3k views ### Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1 Consider an elliptic curve E defined over \mathbb Q. Assume that the rank of E(\mathbb Q) is \geq2. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank = algebraic ... 0answers 2k views ### To what extent does Spec R determine Spec of the Witt vector ring over R? Let R be a perfect \mathbb{F}_p-algebra and write W(R) for the Witt ring [i.e., ring of Witt vectors -- PLC] on R. I want to know how much we can deduce about \text{Spec } W(R) from ... 0answers 2k views ### Grothendieck's Period Conjecture and the missing p-adic Hodge Theories Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over \mathbb Q to \mathbb Q-vectors spaces. They come with the ... 0answers 2k views ### What does the theta divisor of a number field know about its arithmetic? This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link). Let me first ... 0answers 1k views ### What are the potential applications of perfectoid spaces to homotopy theory? This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about ... 0answers 1k views ### A three-line proof of global class field theory? There is an idea (I think originally due to Tate) that class field theory is fundamentally a consequence of Pontrjagin duality and Hilbert Theorem 90. I'm curious whether this can phrased using modern ... 0answers 1k views ### Peano Arithmetic and the Field of Rationals In 1949 Julia Robinson showed the undecidability of the first order theory of the field of rationals by demonstrating that the set of natural numbers \Bbb{N} is first order definable in (\Bbb{Q}, +,... 0answers 632 views ### Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function? The adèles \mathbb A arise naturally when considering the Berkovich space \mathcal M(\mathbb Z) of the integers. Namely, they are the stalk \mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p ... 0answers 4k views ### A generalisation of the equation n = ab + ac + bc In a result I am currently studying (completely unrelated to number theory) I had to examine the solvability of the equation n = ab+ac+bc where n,a,b,c are positive integers 0 < a < b < ... 0answers 798 views ### On certain representations of algebraic numbers in terms of trigonometric functions Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ... 0answers 631 views ### Is there an Ehrhart polynomial for Gaussian integers Let N be a positive integer and let P \subset \mathbb{C} be a polygon whose vertices are of the form (a_1+b_1 i)/N, (a_2+b_2 i)/N, ..., (a_r+b_r i)/N, with a_j + b_j i being various ... 0answers 1k views ### Derivative of Class number of real quadratic fields Let \Delta be a fundamental quadratic discriminant, set N = |\Delta|, and define the Fekete polynomials$$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$Define$$ f_N(X) = \frac{F_N(X)}{... 0answers 703 views ### Is there any positive integer sequence$c_{n+1}=\frac{c_n(c_n+n+d)}n$? In a recent answer Max Alekseyev provided two recurrences of the form mentioned in the title which stay integer for a long time. However, they eventually fail. QUESTION Is there any (added: ... 0answers 949 views ### Enriques surfaces over$\mathbb Z$Does there exist a smooth proper morphism$E \to \operatorname{Spec} \mathbb Z$whose fibers are Enriques surfaces? By a theorem of, independently, Fontaine and Abrashkin, combined with the Enriques-... 0answers 598 views ### Is the Flajolet-Martin constant irrational? Is it transcendental? Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm. In the ... 0answers 2k views ### What are the possible singular fibers of an elliptic fibration over a higher dimensional base? An elliptic fibration is a proper morphism$Y\rightarrow B$between varieties such that the fiber over a general point of the base$B$is a smooth curve of genus one. It is often required for the ... 0answers 744 views ### How to prove the identity$L(2,(\frac{\cdot}3))=\frac2{15}\sum\limits_{k=1}^\infty\frac{48^k}{k(2k-1)\binom{4k}{2k}\binom{2k}k}$? For the Dirichlet character$\chi(a)=(\frac a3)$(which is the Legendre symbol), we have $$L(2,\chi)=\sum_{n=1}^\infty\frac{(\frac n3)}{n^2}=0.781302412896486296867187429624\ldots.$$ Note that this ... 0answers 1k views ### Orders in number fields Let$K$be a degree$n$extension of${\mathbb Q}$with ring of integers$R$. An order in$K$is a subring with identity of$R$which is a${\mathbb Z}$-module of rank$n$. Question: Let$p$be an ... 0answers 589 views ### A modern perspective on the relationship between Drinfeld modules and shtukas Shtukas were defined by Drinfeld as a generalization of Drinfeld modules. While the relationship between the definitions of Drinfeld modules and shtukas is not obvious, one does have a natural ... 0answers 1k views ### Exotic 4-spheres and the Tate-Shafarevich Group The title is a talk given by Sir M. Atiyah in a conference with the following abstract: I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)... 0answers 740 views ### Nekrasov-Okounkov hook length formula I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls the Nekrasov-... 0answers 832 views ### Non-linear expanders? Recall that a family of graphs (indexed by an infinite set, such as the primes, say) is called an expander family if there is a$\delta>0$such that, on every graph in the family, the discrete ... 0answers 1k views ### Defining$\mathbb{Z}$in$\mathbb{Q}$It was proved by Poonen that$\mathbb{Z}$is definable in the structure$(\mathbb{Q}, +, \cdot, 0, 1)$using$\forall \exists$formula. Koenigsmann has shown that$\mathbb{Z}$is in fact definable by ... 0answers 657 views ### Which sets of roots of unity give a polynomial with nonnegative coefficients? The question in brief: When does a subset$S$of the complex$n$th roots of unity have the property that $$\prod_{\alpha\, \in \,S} (z-\alpha)$$ gives a polynomial in$\mathbb R[z]$with ... 0answers 760 views ### Base change for$\sqrt{2}.$This is a direct follow-up to Conjecture on irrational algebraic numbers. Take the decimal expansion for$\sqrt{2},$but now think of it as the base$11$expansion of some number$\theta_{11}.$Is ... 0answers 625 views ### probability of zero subset sum Almost 17 years ago, I asked the following question on USENET, motivated by a method in numerology (I kid you not). Pick integers$n \ge 2$,$k \ge 1$. Toss$nk$-sided dice. The sides of each die ... 0answers 866 views ### Linking formulas by Euler, Pólya, Nekrasov-Okounkov Consider the formal product $$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$ (a) If$z=2$then on the one hand we get Euler's $$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$ on the ... 0answers 1k views ### K-theory and rings of integers From the works of Borel and Quillen there is a connection between the$K$-theory of the ring of integers$\mathfrak{o}_K$in a number field$K$and the arithmetic of the number field. In fact, it is ... 0answers 959 views ### Fake CM elliptic curves Suppose one has an elliptic curve$E$over$\mathbb{Q}$with conductor$N < k^3$for some (large) positive$k$, with the property that its Fourier coefficients satisfy $$a_p=0, \; \mbox{ for all }... 0answers 705 views ### Smooth proper schemes over Z with points everywhere locally This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question. Question. Is there a smooth proper scheme X\to\operatorname{... 0answers 292 views ### Can we write each positive rational number as \frac1{p_1-1}+\ldots+\frac1{p_k-1} with p_1,\ldots,p_k distinct primes? It is well-known that any positive rational number can be written as the sum of finitely many distinct unit fractions. This is easy since$$\frac1n=\frac1{n+1}+\frac1{n(n+1)}\quad\text{for all}\ n=1,2,... 0answers 718 views ### Eichler-Shimura over Totally Real Fields By Eichler-Shimura over totally real fields I mean the following conjecture. Conjecture. Let$K$be a totally real field. Let$f$be a Hilbert eigenform with rational eigenvalues, of parallel weight$...
Assume that you have proved the two inequalities of class field theory, and that you want to show that the Hilbert class field, i.e., the maximal unramified abelian extension, of a number field $K$ ...