On a Conjecture of Sharifi and Mazur’s Eisenstein Ideal

Let $N$ and $p$ be prime numbers $\geq 5$ such that $p$ divides $N-1$. Let $I$ be Mazur’s Eisenstein ideal of level $N$ and $H_+$ be the plus part of $H_1(X_0(N), \mathbf Z_{p})$ for the complex conjugation. We give a conjectural explicit description of the group $I\cdot H_+/I^2\cdot H_+$ in terms of the 2nd $K$-group of the cyclotomic field $\mathbf Q(\zeta _N)$. We prove that this conjecture follows from a conjecture of Sharifi about some Eisenstein ideal of level $\Gamma _1(N)$. Following the work of Fukaya–Kato, we prove partial results on Sharifi’s conjecture. This allows us to prove partial results on our conjecture.

Vanishing theorems for the mod p cohomology of some simple Shimura varieties

We show that the mod p cohomology of a simple Shimura variety treated in Harris-Taylor’s book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree under a mild additional assumption.

Quadratic Twists of Abelian Varieties With Real Multiplication

Let $F$ be a totally real number field and $A/F$ an abelian variety with real multiplication (RM) by the ring of integers $\mathcal {O}$ of a totally real field. Assuming $A$ admits an $\mathcal {O}$-linear 3-isogeny over $F$, we prove that a positive proportion of the quadratic twists $A_d$ have rank 0. If moreover $A$ is principally polarized and $III(A_d)$ is finite, then a positive proportion of $A_d$ have $\mathcal {O}$-rank $1$. Our proofs make use of the geometry-of-numbers methods from our previous work with Bhargava, Klagsbrun, and Lemke Oliver and develop them further in the case of RM. We quantify these results for $A/\mathbb {Q}$ of prime level, using Mazur’s study of the Eisenstein ideal. For example, suppose $p \equiv 10$ or $19 \pmod {27}$, and let $A$ be the unique optimal quotient of $J_0(p)$ with a rational point $P$ of order 3. We prove that at least $25\%$ of twists $A_d$ have rank 0 and the average $\mathcal {O}$-rank of $A_d(F)$ is at most 7/6. Using the presence of two different 3-isogenies in this case, we also prove that roughly $1/8$ of twists of the quotient $A/\langle P\rangle$ have nontrivial 3-torsion in their Tate–Shafarevich groups.

Congruences with Eisenstein series and -invariants

We study the variation of $\unicode[STIX]{x1D707}$-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 $\unicode[STIX]{x1D707}$-invariants to be unbounded along the family, and 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. $\unicode[STIX]{x1D706}=0$). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

On the -invariant of the cyclotomic derivative of a Katzp-adic -function

When the branch character has root number$- 1$, the corresponding anticyclotomic Katz$p$-adic$L$-function vanishes identically. For this case, we determine the$\mu $-invariant of the cyclotomic derivative of the Katz$p$-adic$L$-function. The result proves, as an application, the non-vanishing of the anticyclotomic regulator of a self-dual CM modular form with root number$- 1$. The result also plays a crucial role in the recent work of Hsieh on the Eisenstein ideal approach to a one-sided divisibility of the CM main conjecture.