Comparing local constants of elliptic curves in dihedral extensions

CM椭圆曲线的$L$-函数与广义超几何函数

$L$-function of CM elliptic curves and generalized hypergeometric functions

本研究通过比较动机上同调群中的Bloch元与Ross元，建立了与费马曲线相关的CM椭圆曲线$L$-函数特殊值和广义超几何函数特殊值的关联，为数论中$L$-函数特殊值计算提供新途径。

辛点与日田 - 兰金p进L函数

Plectic points and Hida-Rankin p-adic L-functions

研究通过定义两个变量的反分圆p进L函数，其限制到权重空间可得日田 - 兰金p进L函数。当函数族经过特定模椭圆曲线相关的超收敛模符号时，可得到用辛点计算此类L函数高阶导数的p进格罗斯 - 扎吉尔公式。

Growth of Selmer rank in nonabelian extensions of number fields

Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for rk_p(E/F), generalizing the results in [MR], which applied to dihedral extensions. If K is the (unique) quadratic extension of k in F, G = Gal(F/K), G^+ is the subgroup of elements of G commuting with a choice of involution of F over k, and rk_p(E/K) is odd, then we show that (under mild hypotheses) rk_p(E/F) \ge [G:G^+]$. As a very specific example of this, suppose…

Brauer关系、同源与秩的奇偶性

Brauer relations, isogenies and parities of ranks

本研究提出结合Brauer关系、曲线伽罗瓦覆叠与调节常数理论的统一框架，可重构经典同源、推导塞尔默秩奇偶性局部表达式，涵盖多篇文献中的相关结果。

Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes

We consider the Kolyvagin cohomology classes associated to an elliptic curve $E$ defined over $\mathbb{Q}$ from a computational point of view. We explain how to go from a model of a class as an element of $(E(L)/pE(L))^{\mathrm{Gal}(L/\mathbb{Q})}$, where $p$ is prime and $L$ is a dihedral extension of $\mathbb{Q}$ of degree $2p$, to a geometric model as a genus one curve embedded in $\mathbb{P}^{p-1}$. We adapt the existing methods to compute Heegner points to our situation, and explicitly compute them as elements of $E(L)$. Finally, we compute explicit equations for several genus one curves …

Combinatorial Aspects of Elliptic Curves II: Relationship between Elliptic Curves and Chip-Firing Games on Graphs

Let q be a power of a prime and E be an elliptic curve defined over F_q. In "Combinatorial aspects of elliptic curves" [17], the present author examined a sequence of polynomials which express the N_k's, the number of points on E over the field extensions F_{q^k}, in terms of the parameters q and N_1 = #E(F_q). These polynomials have integral coefficients which alternate in sign, and a combinatorial interpretation in terms of spanning trees of wheel graphs. In this sequel, we explore further ramifications of this connection. In particular, we highlight a relationship between elliptic curves an…

Visualising Sha[2] in Abelian Surfaces

Given an elliptic curve E1 over a number field and an element s in its 2-Selmer group, we give two different ways to construct infinitely many Abelian surfaces A such that the homogeneous space representing s occurs as a fibre of A over another elliptic curve E2. We show that by comparing the 2-Selmer groups of E1, E2 and A, we can obtain information about Sha(E1/K)[2] and we give examples where we use this to obtain a sharp bound on the Mordell-Weil rank of an elliptic curve. As a tool, we give a precise description of the m-Selmer group of an Abelian surface A that is m-isogenous to a produc…