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

Plectic Stark–Heegner points

Plus/minus p-adic L-functions for Hilbert modular forms

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

Let K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

We give archimedean and non-archimedean constructions of Darmon points on modular abelian varieties attached to automorphic forms over arbitrary number fields and possibly non-trivial central character. An effort is made to present a unifying point of view, emphasizing the automorphic nature of the construction.

Let Γ \Gamma be a congruence subgroup of type ( N 1 , N 2 ) (N_1,N_2) and of level N N . We study congruences between weight 2 normalized newforms f f and Eisenstein series E E on Γ \Gamma modulo a prime p \mathfrak {p} above a rational prime p p . Assume that p ∤ 6 N p\nmid 6N , E E is a common eigenfunction for all Hecke operators and f f is ordinary at p \mathfrak {p} . We show that the abelian variety associated to f f and the cuspidal subgroup associated to E E intersect non-trivially in their p p -torsion points. Let A A be a modular elliptic curve over Q \mathbb {Q} with good ordinary reduction at p p . We apply the above result to show that an isogeny of degree divisible by p p from the optimal curve A 1 A_1 in the Q \mathbb {Q} -isogeny class of elliptic curves containing A A to A A extends to an étale morphism of Néron models over Z p \mathbb {Z}_p if p > 7 p>7 . We use this to show that p p -adic distributions associated to the p p -adic L L -functions of A A are Z p \mathbb {Z}_p -valued.

We study generalised Heegner cycles, originally introduced by Bertolini–Darmon–Prasanna for modular curves in Bertolini et al. (Duke Math J 162(6):1033–1148, 2013), in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic p-adic L-function attached to a Coleman family $$f_\infty $$ and an imaginary quadratic field K, constructed in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014). While in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014) only the restriction to the central critical line of this 2 variable p-adic L-function is considered, our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic p-adic L-function restricted to non necessarily central critical lines as a combination of the image of generalized Heegner cycles under a p-adic Abel–Jacobi map. In studying generalised Heegner cycles in the context of Mumford curves, we also obtain an extension of a result of Masdeu (Compos Math 148(4):1003–1032, 2012) for the (one variable) anticyclotomic p-adic L-function of a modular form f and K at non-central critical integers.

On the modularity of supersingular elliptic curves over certain totally real number fields

This article generalizes the geometric quadratic Chabauty method, initiated over Q \mathbb {Q} by Edixhoven and Lido, to curves defined over arbitrary number fields. The main result is a conditional bound on the number of rational points on curves that satisfy an additional Chabauty type condition on the Mordell–Weil rank of the Jacobian. The method gives a more direct approach to the generalization by Dogra of the quadratic Chabauty method to arbitrary number fields.

On the Waldspurger formula and the metaplectic Ramanujan conjecture over number fields

In this paper we generalize, to an arbitrary number field, the theorem which gives the probability that two integers are relatively prime. The probability that two integers are relatively prime is 1/ζ(2), where ζ is the Riemann zeta function and 1/ζ(2)=6/π 2. The theorem for an arbitrary number field states that the probability that two ideals are relatively prime is the reciprocal of the zeta function of the number field evaluated at two. In particular, since the Gaussian integers are an unique factorization domain, we get the probability that two Gaussian integers are relatively prime is 1/ζ G (2) where ζ G is the zeta function associated with the Gaussian integers. In order to calculate the Gaussian probability, we use a theorem that enables us to factor the zeta function into a product of the Riemann zeta function and a Dirichlet series called an L-series. For the Gaussian integers we get: ζ G (2)=ζ(2)L(2,χ), where $$L\left( {2,\chi } \right) = 1 - \frac{1}{{3^2 }} + \frac{1}{{5^2 }} - \frac{1}{{7^2 }} + \cdots .$$ We use this factorization to approximate the Gaussian probability to 17 decimal places.

Injectivity of the specialization homomorphism of elliptic curves

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper E"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an elliptic curve defined over a number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper K slash normal upper F"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">K</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">F</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {K}/\mathrm {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a quadratic extension. For a point <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P element-of normal upper E left-parenthesis normal upper F right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">E</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P\in \mathrm {E}(\mathrm {F})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is a local trace for every completion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper K slash normal upper F"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">K</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">F</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {K}/\mathrm {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we find necessary and sufficient conditions for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to lie in the image of the global trace map. These conditions can then be used to determine whether a quadratic twist of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper E"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , as a genus one curve, has rational points. In the case of quadratic twists of genus one modular curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0 left-parenthesis upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X_0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with squarefree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the existence of rational points corresponds to the existence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -curves of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper K"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

The arithmetic theory of elliptic curves enters the new century with some of its major secrets intact. Most notably, the Birch and Swinnerton-Dyer conjecture, which relates the arithmetic of an elliptic curve to the analytic behaviour of its associated L-series, is still unproved in spite of important advances in the last decades, some of which are recalled in Chapter 1.

abstract: We construct $p$-adic $L$-functions for torsion classes for $\GL(n+1)\times\GL(n)$ and along the way prove new congruences between special values of Rankin-Selberg $L$-functions for $\GL(n+1)\times\GL(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\GL(n\!+\!1)\linebreak\times\GL(n)$. As an application, we establish generic non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.