We introduce a new method for studying mod-ℓ congruences between eigenforms through the modular representation theory of PGL_2(𝔽_p). When p≡± 1 ℓ, we use this theory to construct and describe extra structures on spaces of modular forms with Γ_0(p^2)-level at p and a fixed mod-ℓ Galois representation. The structural results we obtain can be viewed as a refinement of classical level-raising theorems since they not only allow us to prove the existence of congruences, but also to count the number of such congruences. Our methods work equally well in the residually irreducible and residually reducible cases, allowing us to prove several new instances of congruences between Eisenstein series and cuspforms (as well as independently rederiving classical results of Mazur and more recent results of Lang–Wake). Notably, our approach proves these results without computing constant terms of Eisenstein series, without Galois deformation theory and R=𝕋 theorems, and without the Jacquet–Langlands correspondence.

We study the vanishing of $L(f,\chi,j)$ as $f$ runs through all classical forms in a $p$-adic Hida family (including forms with arbitrarily high nebentype at $p$), $\chi$ runs through all characters of $p$-power conductor, and $j$ is a critical value. We show that if infinitely many of these $L$-values vanish (apart from the ones forced to vanish by the sign of their functional equation) then this infinitude of vanishing must be exceptionally regular, so regular in fact that one can typically rule out this possibility in any given example. Indeed, we systematically verified that such regular vanishing does not occur in multiple Hida families twisted by a wide range of quadratic characters by computing the corresponding two-variable $p$-adic $L$-functions via overconvergent modular symbols.

We give an algorithm to determine factorization types of primes in the number fields generated by a single point of odd order on an elliptic curve. We apply this to compute coefficients of the Dedekind zeta function of the field.

We study the variation of mu -invariants of modular forms in a cuspidal Hida family in the case that the family intersects an Eisenstein family. We allow for intersections that occur because of "trivial zeros" (that is, because p divides an Euler factor) as in Mazur's Eisenstein ideal paper, and pay special attention to the case of the 5-adic family passing through the elliptic curve X-0(11).

Let E be an elliptic curve with good ordinary reduction at an odd prime p. Assuming that Greenberg's μ=0 conjecture holds, we show that the λ-invariants of the Mazur–Tate elements attached to E either stabilise to the λ-invariant of the p-adic L-function or they attain the largest possible value at all finite levels. We characterise the latter phenomenon: it occurs if and only if _p(L(E',1)/Ω_E') is negative for some E' that is isogenous to E. Furthermore, we relate this condition to congruences with boundary symbols coming from Eisenstein series. We also study the extension of these results to Hecke eigenforms of weight two.

We prove that the Mazur-Tate elements of an eigenform $f$ sit inside the Fitting ideals of the corresponding dual Selmer groups along the cyclotomic $\mathbb Z_p$-extension (up to scaling by a single constant). Our method begins with the construction of local cohomology classes built via the $p$-adic local Langlands correspondence. From these classes, we build algebraic analogues of the Mazur-Tate elements which we directly verify sit in the appropriate Fitting ideals. Using Kato's Euler system and explicit reciprocity laws, we prove that these algebraic elements divide the corresponding Mazur-Tate elements, implying our theorem.

This article explains how to practically compute L-invariants of p-new eigenforms using p-adic L-series and exceptional zero phenomena. As proof of the utility, we compiled a data set consisting of over 150,000 L-invariants. We analyze qualitative and quantitative features found in the data. This includes conjecturing a statistical law for the distribution of the valuations of L-invariants in a fixed level as the weights of eigenforms approach infinity. One novel point of our investigation is that the algorithm is sensitive to compiling data for fixed Galois representations modulo p. Therefore, we explain new perspectives on L-invariants that are related to Galois representations. We propose understanding the structures in our data through the lens of deformation rings and moduli stacks of Galois representations.

We investigate the λ-invariants of Mazur–Tate elements of elliptic curves defined over the field of rational numbers at primes of additive reduction. We explain their growth and how these invariants relate to other better understood invariants depending on the potential reduction type. We give examples and a conjecture for the additive potentially supersingular case, supported by computational data from Sage in this setting. Further, we extend our results to λ-invariants of Mazur–Tate elements of cuspidal Hecke eigenforms associated with potentially ordinary p-adic Galois representations.

The ghost conjecture, formulated by this article’s authors, predicts the list of p p -adic valuations of the non-zero a p a_p -eigenvalues (“slopes”) for overconvergent p p -adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the “ghost series”). The prediction is restricted to eigenforms whose Galois representation modulo p p is reducible on a decomposition group at p p . It has been discovered, however, that the conjecture is not formulated correctly. Here we explain the issue and propose a salvage.

We study the variation of mu-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the p-adic zeta function. This lower bound forces these mu-invariants to be unbounded along the family, and moreover, we conjecture that this lower bound is an equality. When U_p-1 generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the p-adic L-function is simply a power of p up to a unit (i.e. lambda=0). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".

In a previous article, we constructed an entire power series over $p$-adic weight space (the 'ghost series') and conjectured, in the $\Gamma_0(N)$-regular case, that this series encodes the slopes of overconvergent modular forms of any $p$-adic weight. In this paper, we construct 'abstract ghost series' which can be associated to various natural subspaces of overconvergent modular forms. This abstraction allows us to generalize our conjecture to, for example, the case of slopes of overconvergent modular forms with a fixed residual representation that is locally reducible at $p$. Ample numerical evidence is given for this new conjecture. Further, we prove that the slopes computed by any abstract ghost series satisfy a distributional result at classical weights (consistent with conjectures of Gouv\^ea) while the slopes form unions of arithmetic progressions at all weights not in $\mathbf{Z}_p$.

Let $p$ be an odd prime. Given an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-D_K})$ where $p$ splits with $D_K>3$, and a $p$-ordinary newform $f \in S_k(\Gamma_0(N))$ such that $N$ verifies the Heegner hypothesis relative to $K$, we prove a $p$-adic Gross-Zagier formula for the critical slope $p$-stabilization of $f$ (assuming that it is non-$\theta$-critical). In the particular case when $f=f_A$ is the newform of weight $2$ associated to an elliptic curve $A$ that has good ordinary reduction at $p$, this allows us to verify a conjecture of Perrin-Riou. The $p$-adic Gross-Zagier formula we prove has applications also towards the Birch and Swinnerton-Dyer formula for elliptic curves of analytic rank one.

We give a sufficient condition, namely “Buzzard irregularity”, for there to exist a cuspidal eigenform which does not have integral p-adic slope.

We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".

We establish the freeness of certain anticyclotomic Selmer groups of modular forms. The freeness of these Selmer groups plays a key role in the Euler system arguments introduced by Bertolini and Darmon in their work on the anticyclotomic main conjecture for modular forms. In particular, our result fills some implicit gaps which appeared in generalizations of the Bertolini-Darmon result to the case where the associated residual representation is not minimally ramified. The removal of such a minimal ramification hypothesis is essential for applications involving congruences of modular forms.

In Pollack and Stevens (Ann Sci Éc Norm Supér 44(1):1–42, 2011), efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of p-adic L-functions and have further been applied to compute rational points on elliptic curves (e.g. Darmon and Pollack in Israel J Math 153:319–354, 2006, Trifkovic in Duke Math J 135(3):415–453, 2006). In this paper, we generalize these algorithms to the case of families of overconvergent modular symbols. As a consequence, we can compute p-adic families of Hecke-eigenvalues, two-variable p-adic L-functions, L-invariants, as well as the shape and structure of ordinary Hida–Hecke algebras.

We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms 'near the boundary' of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the p operator never vanish modulo <p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.

In this paper, we give an introduction to the theory of overconvergent modular symbols and their connection to p-adic L-functions. Alongside the theory, we describe many explicit examples to illustrate the concrete nature of this subject, and give algorithms for computing inside of these infinite-dimensional spaces.