For multiplicative characters of a non-Archimedean local field, we have a formula for epsilon factors due to John Tate. Before Tate, Erich Lamprecht also gave a formula for local epsilon factors of linear characters. Then Tate generalizes the formula for epsilon factors. In this paper, we give a very short and neat proof of the Lamprecht-Tate formula. We also show that the famous twisting formula of Deligne is a special case of the Lamprecht-Tate formula.
We use the invariant theory of binary quartics to give a new formula for the Cassels-Tate pairing on the $2$-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role in our construction is played by a certain $K3$ surface defined by a $(2,2,2)$-form.
In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption that all points in $J[2]$ are $K$-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating we can indeed improve the rank bound given by a 2-descent via computing the Cassels-Tate pairing.
Explicit formulas are proved for the $5$-torsion points on the Tate normal form $E_5$ of an elliptic curve having $(X,Y)=(0,0)$ as a point of order $5$. These formulas express the coordinates of points in $E_5[5] - \langle(0,0)\rangle$ as products of linear fractional quantities in terms of $5$-th roots of unity and a parameter $u$, where the parameter $b$ which defines the curve $E_5$ is given as $b=(\varepsilon^5 u^5- \varepsilon^{-5})/(u^5+1)$ and $\varepsilon = (-1+\sqrt{5})/2$. If $r(τ)$ is the Rogers-Ramanujan continued fraction and $b=r^5(τ)$, then the coordinates of points of order $5$…
We give an explicit formula for the arithmetic intersection number of CM cycles on Lubin-Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions $K_1,K_2/F$ of non-Archimedean local fields $F$. Our formula works for all cases, $K_1$ and $K_2$ can be either the same or different, ramify or unramified. As applications, this formula translate the linear Arithmetic Fundamental Lemma (linear AFL) into a comparison of integrals. This formula can also be used to recover …
We prove a formula, which, given a principally polarized abelian variety $(A,λ)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron--Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,λ)$.
We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate-Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.
It is shown that the number of irreducible quartic factors of the form $g(x) = x^4+ax^3+(11a+2)x^2-ax+1$ which divide the Hasse invariant of the Tate normal form $E_5$ in characteristic $l$ is a simple linear function of the class number $h(-5l)$ of the field $\mathbb{Q}(\sqrt{-5l})$, when $l \equiv 2,3$ modulo $5$. A similar result holds for irreducible quadratic factors of $g(x)$, when $l \equiv 1, 4$ modulo $5$. This implies a formula for the number of linear factors over $\mathbb{F}_p$ of the supersingular polynomial $ss_p^{(5*)}(x)$ corresponding to the Fricke group $Γ_0^*(5)$.
In arXiv:1208.1945, Shin and Templier proved certain equidistribution bounds on local components of certain families of automorphic representations. We extend their weight-aspect results to families of automorphic representations where the Archimedean component is restricted to a single discrete-series representation instead of an entire $L$-packet. We do this by using a so-called "hyperendoscopy" version of the stable trace formula developed by Ferrari. The main technical difficulties are defining a version of hyperendoscopy that works for groups without simply connected derived subgroup and …
We develop a descent criterion for $K$-linear abelian categories. Using recent advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti-Tate groups on complete curves over a finite field. We conjecture that such Barsotti-Tate groups "come from" a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over $\mathbb{F}_q$. Along the way, we formulate a conjecture on the field-of-coefficients of certain compatible systems.
Following the natural instinct that when a group operates on a number field then every term in the class number formula should factorize `compatibly' according to the representation theory (both complex and modular) of the group, we are led to some natural questions about the $p$-part of the classgroup of any CM Galois extension of $\Q$ as a module for $\Gal(K/Q)$, in the spirit of Herbrand-Ribet's theorem on the $p$-component of the class number of $Q(ζ_p)$. In trying to formulate these questions, we are naturally led to consider $L(0,ρ)$, for $ρ$ an Artin representation, in situations where …
A formula is proved for the number of linear factors over $\mathbb{F}_l$ of the Hasse invariant of the Tate normal form $E_5(b)$ for a point of order $5$, as a polynomial in the parameter $b$, in terms of the class number of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-l})$, proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term $-1$, and a theorem is stated for the number of quartic factors of a specific form in terms of the class number of $\mathbb{Q}(\sqrt{-5l})$. These results are shown to imply a recent conjecture of Nakaya…
Given a number field $K$ with a Hecke character $χ$, for each place $ν$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $χ$. These theories appear in the study of $p$-adic string theory and $p$-adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of $χ$. We find that the Green's function is given by the local functional equation for Zeta integrals. Furthermore, considering all places $ν$, the field theory t…
In this paper we study the derivatives of Frobenius and the derivatives of Hodge-Tate weights for families of Galois representations with triangulations. We give a generalization of the Fontaine-Mazur L-invariant and use it to build a formula which is a generalization of the Colmez-Greenberg-Stevens formula.
In these notes we define and study the Shahidi local coefficients matrix associated with a genuine principal series representation I(σ) of an n-fold cover of p-adic SL(2,F) and an additive character ψ. The conjugacy class of this matrix is an invariant of the inducing representation σ and ψ and its entries are linear combinations of Tate or Tate type γ-factors. We relate these entries to functional equations associated with linear maps defined on the dual of the space of Schwartz functions. As an application we give new formulas for the Plancherel measure and use these to relate principal seri…
