Split Multiplicative Reduction Research Articles

Watkins’s conjecture for elliptic curves with non-split multiplicative reduction

Let E E be an elliptic curve over the rational numbers. Watkins [Experiment. Math. 11 (2002), pp. 487–502 (2003)] conjectured that the rank of E E is bounded by the 2 2 -adic valuation of the modular degree of E E . We prove this conjecture for semistable elliptic curves having exactly one rational point of order 2 2 , provided that they have an odd number of primes of non-split multiplicative reduction or no primes of split multiplicative reduction.

Torsion properties of modified diagonal classes on triple products of modular curves

AbstractConsider three normalized cuspidal eigenforms of weight $2$ and prime level p. Under the assumption that the global root number of the associated triple product L-function is $+1$ , we prove that the complex Abel–Jacobi image of the modified diagonal cycle of Gross–Kudla–Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel–Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow–Heegner points associated with modified diagonal cycles on elliptic curves of prime conductor with split multiplicative reduction. The approach also works in the case of composite square-free level.

Tamagawa numbers of elliptic curves with prescribed torsion subgroup or isogeny

We study Tamagawa numbers of elliptic curves with torsion Z/2Z⊕Z/14Z over cubic fields and of elliptic curves with an n-isogeny over Q, for n∈{6,8,10,12,14,16,17,18,19,37,43,67,163}. Bruin and Najman [3] proved that every elliptic curve with torsion Z/2Z⊕Z/14Z over a cubic field is a base change of an elliptic curve defined over Q. We find that Tamagawa numbers of elliptic curves defined over Q with torsion Z/2Z⊕Z/14Z over a cubic field are always divisible by 142, with each factor 14 coming from a rational prime with split multiplicative reduction of type I14k, one of which is always p=2. The only exception is the curve 1922.e2, with cE=c2=14. The same curves defined over cubic fields over which they have torsion subgroup Z/2Z⊕Z/14Z turn out to have the Tamagawa number divisible by 143. As for n-isogenies, Tamagawa numbers of elliptic curves with an 18-isogeny must be divisible by 4, while elliptic curves with an n-isogeny for the remaining n from the mentioned set must have Tamagawa numbers divisible by 2, except for finite sets of specified curves.

Zero Cycles on a Product of Elliptic Curves Over a p-adic Field

Abstract We consider a product $X=E_1\times \cdots \times E_d$ of elliptic curves over a finite extension $K$ of ${\mathbb{Q}}_p$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has supersingular reduction. Under these assumptions, we prove that the Albanese kernel of $X$ is the direct sum of a finite group and a divisible group, extending work by Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map $CH_0(X)/p^n\rightarrow H^{2d}_{\acute{\textrm{e}}\textrm{t}}(X, \mu _{p^n}^{\otimes d})$. We give specific criteria that guarantee this map is injective for every $n\geq 1$. When all curves have good ordinary reduction, we show that it suffices to extend to a specific finite extension $L$ of $K$ for these criteria to be satisfied. This extends previous work by Yamazaki and Hiranouchi.

Indecomposability and cyclicity in the p-primary part of the Brauer group of a p-adic curve

We obtain two results about the p-primary part of the Brauer group of a p-adic curve X. First, assuming enough roots of unity and that X has good reduction, we construct indecomposable K(X)-division algebras of period p2 and index p3. Second, for an elliptic curve X with split multiplicative reduction, we show that all order p elements of are -cyclic, and that if moreover X is defined over that all order pr elements of are -cyclic for

A converse to a theorem of Gross, Zagier, and Kolyvagin

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes, then \[ \mathrm{rank}_{\mathbb{Z}} E(\mathbb{Q}) =1\quad \mathrm{and}\quad \#\Sha(E) \lt \infty X(E)\implies \mathrm{ord}_{s=1}L(E,s) = 1. \] We also prove the corresponding result for the abelian variety associated with a weight 2 newform $f$ of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for $f$ and $H^1_f(\mathbb{Q},V)$, where $V$ is the $p$-adic Galois representation associated with $f$, that ensure that $\mathrm{ord}_{s=1}L(f,s) = 1$. The main theorem is proved using the Iwasawa theory of $V$ over an imaginary quadratic field to show that the $p$-adic logarithm of a suitable Heegner point is non-zero.

On the $p$-converse of the Kolyvagin–Gross–Zagier theorem

Let A / \mathbb Q be an elliptic curve having split multiplicative reduction at an odd prime p . Under some mild technical assumptions, we prove the statement: \mathrm {rank}_{\mathbb Z} A(\mathbb Q)=1\ \text{and}\ \# (\mathrm {Ш}(A / \mathbb Q)_{p^{\infty}}\big)<\infty\ \ \implies\ \ \mathrm{ord}_{s=1}L(A / \mathbb Q,s)=1, thus providing a ‘ p -converse’ to a celebrated theorem of Kolyvagin–Gross–Zagier.

Heights of points with bounded ramification

Let $E$ be an elliptic curve defined over a number field $K$ with fixed non-archimedean absolute value $v$ of split-multiplicative reduction, and let $f$ be an associated Latt\`es map. Baker proved in 2003 that the N\'eron-Tate height on $E$ is either zero or bounded from below by a positive constant, for all points of bounded ramification over $v$. In this paper we make this bound effective and prove an analogue result for the canonical height associated to $f$. We also study variations of this result by changing the reduction type of $E$ at $v$. This will lead to examples of fields $F$ such that the N\'eron-Tate height on non-torsion points in $E(F)$ is bounded from below by a positive constant and the height associated to $f$ gets arbitrarily small on $F$. The same example shows, that the existence of such a lower bound for the N\'eron-Tate height is in general not preserved under finite field extensions.

Exceptional zero formulae and a conjecture of Perrin-Riou

Let $$A/\mathbf{Q}$$ be an elliptic curve with split multiplicative reduction at a prime p. We prove (an analogue of) a conjecture of Perrin-Riou, relating p-adic Beilinson–Kato elements to Heegner points in $$A(\mathbf{Q})$$ , and a large part of the rank-one case of the Mazur–Tate–Teitelbaum exceptional zero conjecture for the cyclotomic p-adic L-function of A. More generally, let f be the weight-two newform associated with A, let $$f_{\infty }$$ be the Hida family of f, and let $$L_{p}(f_{\infty },k,s)$$ be the Mazur–Kitagawa two-variable p-adic L-function attached to $$f_{\infty }$$ . We prove a p-adic Gross–Zagier formula, expressing the quadratic term of the Taylor expansion of $$L_{p}(f_{\infty },k,s)$$ at $$(k,s)=(2,1)$$ as a non-zero rational multiple of the extended height-weight of a Heegner point in $$A(\mathbf{Q})$$ .

Akashi series of Selmer groups

AbstractWe study the Selmer group of an elliptic curve over an admissible p-adic Lie extension of a number field F. We give a formula for the Akashi series attached to this module, in terms of the corresponding objects for the cyclotomic ℤp-extension and certain correction terms. This extends our earlier work [16], in particular since it applies to elliptic curves having split multiplicative reduction at some primes above p, in which case the Akashi series can have additional zeros.

Zero-cycles on products of elliptic curves over p-adic fields

The structure of CH0(X)/p is determined when X is a product of two elliptic curves over a p-adic field which have ordinary good reduction or split multiplicative reduction.

Curves with prescribed period and index over local fields

The index of a curve is the smallest positive degree of divisors which are rational over a fixed base field. The period is the smallest positive degree of divisor classes rational over the base field. Lichtenbaum proved certain divisibility conditions relating the period, index, and genus of a curve over a local field. We prove that his conditions are sharp by finding a curve of every admissible period, index, and genus. Our proof constructs degree two covers of elliptic curves with split multiplicative reduction.

Teitelbaum's exceptional zero conjecture in the function field case

The exceptional zero conjecture relates the first derivative of the $p$-adic $L$-function of a rational elliptic curve with split multiplicative reduction at $p$ to its complex $L$-function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field $\FQ(T)$ with split multiplicative reduction at two places $\fp$ and $\infty$, avoiding the construction of a $\fp$-adic $L$-function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals over arbitrary function fields. A function field version of Darmon's period conjecture is also obtained.

Non-Archimedean Integration and Elliptic Curves over Function Fields

Let F be a global function field of characteristic p and E/F an elliptic curve with split multiplicative reduction at the place ∞: then E can be obtained as a factor of the Jacobian of some Drinfeld modular curve. This fact is used to associate to E a measure μE on P1(F∞). By choosing an appropriate embedding of a quadratic unramified extension K/F into the matrix algebra M2(F), μE is pushed forward to a measure on a p-adic group G, isomorphic to an anticyclotomic Galois group over the Hilbert class field of K. Integration on G then yields a Heegner point on E when ∞ is inert in K and an analogue of the L-invariant if ∞ is split. In the last section, the same methods are extended to integration on a geometric cyclotomic Galois group.

Local heights on abelian varieties with split multiplicative reduction

We express Neron functions and Schneider's local p-adic height pairing on an abelian variety A with split multiplicative reduction with theta functions and their automorphy factors on the rigid analytic torus uniformizing A.Moreover, we show formulas for the ρ-splittingsof the Poincare biextension corresponding to Neron's and Schneider's local height pairings.

P-adic periods and modular symbols of elliptic curves of prime conductor

Under certain assumptions, we prove a conjecture of Mazur and Tate describing a relation between the modular symbol attached to an elliptic curve with split multiplicative reduction atp, and itsp-adic period. We generalize this relation to modular forms of weight 2 with coefficients not necessarily in .