Mazur–Tate Elements in Iwasawa Theory

• Mazur–Tate elements are defined as group ring elements constructed from modular symbols and theta elements that interpolate critical L-values associated with modular forms and elliptic curves.
• They link analytical properties of L-functions to the algebraic structure of Selmer groups through conjectures relating Fitting ideals to refined BSD-type invariants.
• Advanced Iwasawa theory employs Mazur–Tate elements to study μ- and λ-invariants, congruences among forms, and growth patterns in cyclotomic and anticyclotomic extensions.

Mazur–Tate elements are group ring-valued arithmetic objects associated with modular forms and elliptic curves that encode deep information about special values of $L$-functions, Selmer groups, Iwasawa theory, and the refined arithmetic of elliptic curves over number fields and their towers of extensions. Originally introduced in work of Mazur and Tate for the formulation of refined Birch and Swinnerton-Dyer (BSD) type conjectures, they generalize Stickelberger elements and interpolate critical $L$-values in the group algebras of Galois extensions. In various settings (cyclotomic, anticyclotomic, higher weight, non-ordinary primes), these elements play a unifying role as the “analytic” side in Iwasawa-theoretic main conjectures relating special $L$-values to Fitting and characteristic ideals of Selmer groups.

1. Construction and Definition

Mazur–Tate elements are constructed, for a modular form $f$ or an elliptic curve $E$, as elements in the group ring $\mathcal{O}[G]$—where $G$ is typically the Galois group of a finite abelian extension (such as the $n$-th layer in a $\mathbb{Z}_p$-extension or the Galois group of a ring class field over an imaginary quadratic field)—via modular symbols or theta elements.

Given a modular symbol $\varphi$ attached to $f$, or the associated modular form $f_E$ for $E$, the Mazur–Tate element $\theta_n(f)$ at finite level $n$ is constructed as

$$\theta_n(f) = \sum_{a \in (\mathbb{Z}/p^n\mathbb{Z})^\times} \varphi \mid \left( \begin{array}{cc} -a & p^n \ 1 & 0 \end{array} \right) \cdot \sigma_a,$$

where $\sigma_a \in G_n$ corresponds to $a$ via the identification of $G_n$ with $(\mathbb{Z}/p^n\mathbb{Z})^\times$ (or an appropriate narrow ray class group).

The crucial interpolation property is that for any character $\chi$ of $G$, the value $\chi(\theta_n(f))$ gives (up to an explicit period) the algebraic part of $L(f, \chi, 1)$. In modular form notation, these elements encode congruences and relations among $L$-values twisted by Dirichlet or ring class characters.

In the context of anticyclotomic towers (over an imaginary quadratic field $K$), theta elements are defined via sums over Heegner or Gross points on Shimura curves, with values given by optimal embeddings or CM points, and can be explicitly described as

$L_p(K(m), f) = \theta_m(f)\cdot \theta_m^*(f)$

in the group algebra $\mathcal{O}[\mathrm{Gal}(K(m)/K)]$ (Kim, 2016).

2. Algebraic and Analytic Interpretations

Mazur–Tate elements have a dual algebraic and analytic role:

• Analytically, they interpolate special (critical) values of complex and $p$-adic $L$-functions of modular forms and their twists, providing group ring elements that generalize the Stickelberger elements for Dirichlet $L$-functions (Ota, 2015).
• Algebraically, they are conjecturally related (sometimes up to explicit correction factors or normalization constants) to Fitting ideals or characteristic ideals of Pontryagin dual Selmer groups. The main refined conjectures state

$$L_p(K(m), f) \in \mathrm{Fitt}_{\mathcal{O}[\mathrm{Gal}(K(m)/K)]}(\mathrm{Sel}(K(m), A_f)^\vee),$$

where $A_f$ is the relevant Galois representation attached to $f$ (Kim, 2016).

In Iwasawa theory, these elements are the “analytic side” of main conjectures relating the size and structure of Selmer groups to (analytic) $p$-adic $L$-functions.

3. Role in Refined BSD-type Conjectures

Refined BSD-type conjectures posit that the order of vanishing of the Mazur–Tate element at the augmentation ideal (the "order part") is determined by arithmetic invariants, specifically the Mordell–Weil rank and the number of split multiplicative primes (or appropriate local correction terms): $\theta_S \in I_S^{r_E + sp(S)},$ where $sp(S)$ counts split multiplicative primes and $r_E$ is the Mordell–Weil rank (Ota, 2015). The leading coefficient (modulo a higher power of the augmentation ideal) is predicted to be a precise formula involving Tamagawa numbers, the regulator, periods, and the order of the Tate–Shafarevich group (Burns et al., 2021). Recent work provides unconditional theoretical evidence for such predictions, especially relating the "order of vanishing" and the leading term to Kato's Euler system and the Bockstein regulator (Burns et al., 2021). Numerical studies using SageMath confirm and refine earlier conjectural formulas, clarifying the significance of torsion subgroups and subtle normalization issues (Llerena-Córdova, 23 Dec 2024).

4. Iwasawa Invariants and Asymptotic Behavior

In the cyclotomic direction, Mazur–Tate elements and their expansions in the Iwasawa algebra enable a paper of their $\mu$- and $\lambda$-invariants, measuring $p$-divisibility and degree of nullity, respectively, in the expansion

$\theta_n(E) = \sum_i a_i (T)^i,$

with $T = \gamma - 1$ (Lei et al., 21 Dec 2024, Gajek-Leonard et al., 14 Aug 2025). Recent results establish explicit asymptotic formulas for the $\lambda$-invariants depending on the reduction type:

• For curves with additive reduction and semistability defect $2$,

$$\lambda(\theta_{n,i}(E)) = \frac{p-1}{2}p^{n-1} + \lambda(E^F, \omega^{(p-1)/2 + i})$$

for the appropriate quadratic twist $E^F$ (Lei et al., 21 Dec 2024).

• For good ordinary primes with reducible mod $p$ representation,

$\lambda(\theta_n(E)) = p^n - 1$

for all $n$ (Lei et al., 21 Dec 2024).

• Non-ordinary modular forms of arbitrary weight $k$ and $p>k-1$ satisfy

$$\lambda(\Theta_{n,j}(f, \omega^i)) = (k-1)q_n + \lambda(f, \star, \omega^i)$$

where $\star$ denotes the sharp/flat decomposition and $q_n$ is a combinatorial factor depending on $n$ (Gajek-Leonard et al., 14 Aug 2025).

In higher weight and non-ordinary situations, sharp/flat decompositions, refined modular symbol techniques, and $p$-adic Hodge-theoretic input are essential for analyzing these invariants.

5. Mazur–Tate Elements and Selmer Groups: Fitting Ideals

The central Iwasawa-theoretic conjectures and theorems assert that (up to explicit normalization), the Mazur–Tate element or its suitable stabilization generates the Fitting ideal of the Pontryagin dual of a (minimal or signed) Selmer group: $$\Theta_{n,m} \in \mathrm{Fitt}_{\Lambda_{n,m}}\left( \mathcal{X}_{\mathrm{Gr}}(K'(\mathfrak{f}), A) \right)$$ for strict Greenberg Selmer groups over finite layers in a $\mathbb{Z}_p^2$-extension (Dion, 23 May 2024). In the anticyclotomic context (especially at supersingular or inert primes), the analogous statement is

$$L_p(E/K_n) \in \mathrm{Fitt}_{\Lambda_n}\left(\mathrm{Sel}_{p^\infty}(E/K_n)^\vee\right)$$

with detailed control of signed or plus/minus Selmer groups via plus/minus theory for supersingular reduction (Kim, 3 Mar 2024, Shii, 12 Mar 2025). Essential technical input comes from the work of Burungale, Büyükkoduk, and Lei resolving local cohomological decompositions required for the plus/minus constructions at inert primes (Shii, 12 Mar 2025).

6. Modularity, Congruences, and Extensions

Mazur–Tate elements are defined via modular symbols, and their arithmetic is highly sensitive to congruences among modular forms. Congruences between forms of different weights, or between modular forms and Eisenstein series, impact the Iwasawa invariants of the associated Mazur–Tate elements (Doyon et al., 2022, Doyon et al., 2021). Multiplicity one results in modular symbol cohomology, especially in the Fontaine–Laffaille ("small weight") range, allow the transfer of invariants between modular forms with congruent residual representations (Gajek-Leonard et al., 14 Aug 2025). In the Rankin–Selberg setting, analogous theta elements and divisibility conjectures have been formulated and proved under technical hypotheses (Cauchi et al., 2020).

Anticyclotomic analogues (including for supersingular and inert primes) have been developed, with analogous statements for Fitting ideals of Selmer groups over ring class field towers, and with explicit reciprocity laws connecting theta elements from Heegner cycles to cohomology classes underlying Euler systems (Kim, 2016, Shii, 12 Mar 2025).

7. Open Problems and Future Directions

Mazur–Tate elements remain a central focus for ongoing research. Key open directions include:

• Proving equality (not just inclusion) in Fitting or characteristic ideals in various settings (especially beyond the “weak” main conjecture) (Kim, 3 Mar 2024, Shii, 12 Mar 2025).
• Extending asymptotic and explicit formulas for Iwasawa invariants beyond current ranges (e.g., outside the Fontaine–Laffaille domain, or for more general residual representations) (Gajek-Leonard et al., 14 Aug 2025).
• Understanding the full implications for the refined BSD formula, specifically the leading term and the order of vanishing components, relating with Euler system and regulator computations (Burns et al., 2021).
• Systematic paper of the impact of congruences and torsion subtleties on the refined conjectural formulas, informed by recent numerical studies (Llerena-Córdova, 23 Dec 2024).
• Adapting Mazur–Tate element machinery to higher weight modular forms, Hilbert modular forms, or more general automorphic settings, and examining analogues for Selmer groups of higher-dimensional motives (Gajek-Leonard et al., 14 Aug 2025).
• Investigation of the interaction with Iwasawa theory in multivariable ($\mathbb{Z}_p^2$ or more general) extensions (Dion, 23 May 2024).

Mazur–Tate elements thus remain foundational in the paper of the deep connections between $L$-values, modularity, Selmer groups, and Iwasawa theoretic invariants for arithmetic geometry.