Stark-Heegner points on modular jacobians

We present a construction which lifts Darmon’s Stark–Heegner points from elliptic curves to certain modular Jacobians. Let N be a positive integer and let p be a prime not dividing N. Our essential idea is to replace the modular symbol attached to an elliptic curve E of conductor Np with the universal modular symbol for Γ0(Np). We then construct a certain torus T over Qp and lattice L ⊂ T, and prove that the quotient T/Lis isogenous to the maximal toric quotient J0(Np) p-new of the Jacobian of X0(Np).This theorem generalizes a conjecture of Mazur, Tate, and Teitelbaum on the p-adic periods of elliptic curves, which was proven by Greenberg and Stevens. As a by-product of our theorem, we obtain an efficient method of calculating the p-adic periods of J0(Np) p-new

Ranks of matrices of logarithms of algebraic numbers I: the theorems of Baker and Waldschmidt-Masser

Let $\mathscr{L}$ denote the $\mathbf{Q}$-vector space of logarithms of algebraic numbers. In this expository work, we provide an introduction to the study of ranks of matrices with coefficients in $\mathscr{L}$. We begin by considering a slightly different question, namely we present a proof of a weak form of Baker's Theorem. This states that a collection of elements of $\mathscr{L}$ that is linearly independent over $\mathbf{Q}$ is in fact linear independent over $\overline{\mathbf{Q}}$. Next we recall Schanuel's Conjecture and prove Ax's analogue of it over $\mathbf{C}((t))$. We then consider arbitrary matrices with coefficients in $\mathscr{L}$ and state the Structural Rank Conjecture, which gives a conjecture for the rank of a general matrix with coefficients in $\mathscr{L}$. We prove the theorem of Waldschmidt and Masser, which provides a lower bound giving a partial result toward the Structural Rank Conjecture. We conclude by stating a new conjecture that we call the Matrix Coefficient Conjecture, which gives a necessary condition for a square matrix with coefficients in $\mathscr{L}$ to be singular.Comment: 45 page

On Constructing Extensions of Residually Isomorphic Characters

This is an exposition of our joint work with Kakde, Silliman, and Wang, in which we prove a version of Ribet's Lemma for $\mathrm{GL}_2$ in the residually indistinguishable case. We suppose we are given a Galois representation taking values in the total ring of fractions of a complete reduced Noetherian local ring $\mathbf{T}$, such that the characteristic polynomial of the representation is reducible modulo some ideal $I \subset \mathbf{T}$. We assume that the two characters that arise are congruent modulo the maximal ideal of $\mathbf{T}$. We construct an associated Galois cohomology class valued in a $\mathbf{T}$-module that is "large" in the sense that its Fitting ideal is contained in $I$. We make some simplifying assumptions that streamline the exposition -- we assume the two characters are actually equal, and we ignore the local conditions needed in arithmetic applications.Comment: 23 page

Brumer-Stark Units and Hilbert's 12th Problem

Let $F$ be a totally real field of degree $n$ and $p$ an odd prime. We prove the $p$-part of the integral Gross-Stark conjecture for the Brumer-Stark $p$-units living in CM abelian extensions of $F$. In previous work, the first author showed that such a result implies an exact $p$-adic analytic formula for these Brumer-Stark units up to a bounded root of unity error, including a "real multiplication" analogue of Shimura's celebrated reciprocity law in the theory of Complex Multiplication. In this paper we show that the Brumer-Stark units, along with $n-1$ other easily described elements (these are simply square roots of certain elements of $F$) generate the maximal abelian extension of $F$. We therefore obtain an unconditional solution to Hilbert's 12th problem for totally real fields, albeit one that involves $p$-adic integration, for infinitely many primes $p$. Our method of proof of the integral Gross-Stark conjecture is a generalization of our previous work on the Brumer-Stark conjecture. We apply Ribet's method in the context of group ring valued Hilbert modular forms. A key new construction here is the definition of a Galois module $\nabla_{\!\mathscr{L}}$ that incorporates an integral version of the Greenberg-Stevens $\mathscr{L}$-invariant into the theory of Ritter-Weiss modules. This allows for the reinterpretation of Gross's conjecture as the vanishing of the Fitting ideal of $\nabla_{\!\mathscr{L}}$. This vanishing is obtained by constructing a quotient of $\nabla_{\!\mathscr{L}}$ whose Fitting ideal vanishes using the Galois representations associated to cuspidal Hilbert modular forms.Comment: 70 page

The p-adic L-functions of Evil Eisenstein Series

We compute the $p$-adic $L$-functions of evil Eisenstein series, showing that they factor as products of two Kubota--Leopoldt $p$-adic $L$-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.Comment: 49 page

On the Brumer-Stark Conjecture

Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element $\Theta^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $\text{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with $\mathbf{Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $\text{Cl}^T(H) \otimes \mathbf{Z}[1/2]$ in terms of Stickelberger elements. We also show that this stronger result implies Rubin's higher rank version of the Brumer-Stark conjecture, again away from 2. Our technique is a generalization of Ribet's method, building upon on our earlier work on the Gross-Stark conjecture. Here we work with group ring valued Hilbert modular forms as introduced by Wiles. A key aspect of our approach is the construction of congruences between cusp forms and Eisenstein series that are stronger than usually expected, arising as shadows of the trivial zeroes of $p$-adic $L$-functions. These stronger congruences are essential to proving that the cohomology classes we construct are unramified at $p$.Comment: 99 pages (A reference is updated in the new version