In this post, the abbreviation "MTC" and "FPdim" stand for "Modular Tensor Category" and "Frobenius-Perron dimension".
From [DLN, Theorem II (iii)], where the modular data is normalized, we get (see Appendix):
Theorem 1: A prime $p$ dividing the global dimension norm of a rank $r$ MTC satisfies $p \leq 2r + 1$.
Here, the norm is defined as the product of the distinct Galois conjugates.
This inequality is optimal since the equality is true for some MTCs (e.g., the Yang-Lee category where $r=2$ and $p=5$), as classified in [NWZ].
However, in the integral case, expectations are higher, as suggested by the following theorem (proved in Appendix also):
Theorem 2: Let $G$ be a finite group. Consider the Drinfeld center $\mathcal{Z}(\operatorname{Rep}(G))$ as an integral MTC. Let $r$ be its rank. For every prime $p$ dividing its ${\rm FPdim}=|G|^2$, then $p \leq r$.
More generally:
Question: In the case of an integral MTC, is it true that $p \le r$?
If this holds, then the inequality is optimal as demonstrated by the example of a pointed MTC of prime rank.
The following was realized from private discussions with Eric Rowell (in July 2022) and Andrew Schopieray (in March 2024), see more details in Appendix:
Theorem 3: For an integral MTC, for every prime $p$ dividing the global ${\rm FPdim}$, there is a basic ${\rm FPdim}$ of multiplicity $m$ such that $p \le 2m+1$.
More generally , we can prove that:
Theorem 4: For an integral MTC, let $S$ be the set of odd prime factors of the global ${\rm FPdim}$ . There is a partition $(S_i)$ of $S$, and multiplicities $(m_i)$ of some distinct basic ${\rm FPdim}$s such that $$m_i \ge \frac{1}{2} {\rm lcm}_{p \in S_i}(p-1).$$
The function $\lambda(n)$, known as the Carmichael function, represents the exponent of the multiplicative group of integers modulo $n$. Therefore, $\lambda(\text{prod}(S_i)) = {\rm lcm}_{p \in S_i}(p-1)$.
Appendix
Galois action on the modular data
Let $(s,t)$ be a normalized modular data. A Galois automorphism $\sigma$ induces a permutation $X \to \sigma(X)$ on the simple objects, and acts as follows on $\dim$, $s$ and $t$:
(1) $ \sigma(\dim(X)^2) = \frac{\sigma(\dim(\mathcal{C}))}{\dim(\mathcal{C})}\dim(\sigma(X))^2$,
(2) $\sigma(s_{X,Y}^2) = s_{X,\sigma(Y)}^2$,
(3) $\sigma^2(t_X) = t_{\sigma(X)}$, see [DLN, Theorem II (iii)].
See for example [PSYZ, Section 2] for an explicit normalization of the modular data.
Proof of Theorem 1:
If $p=2,3$ then $p \le 2r+1$ trivially as $r \ge 1$. Let $p \neq 2,3$ be a prime factor of the global dimension norm. By Cauchy's theorem in [BNRW], $p$ divides ${\rm ord(t)}$. So there must be a simple object $X$ such that $p$ divides the conductor of $t_X$, thus the orbit of $(\sigma^2(t_X))$ has at least $(p-1)/2$ distinct elements, because the group of units in $\mathbb{Z}/p\mathbb{Z}$ is cyclic of order $p-1$, so it has an element $g$ with ${\rm ord}(g^2) = (p-1)/2$. So by (3), $r \ge (p-1)/2$, i.e., $p \le 2r+1$. $\square$
Proof of Theorem 3
Consider the the orbit $(\sigma^2(t_X))$ with at least $(p-1)/2$ distinct elements from the proof of Theorem 1. By (3), the orbit $(\sigma(X))$ has also at least $(p-1)/2$ distinct elements. By applying (1) on the (weakly) integral case, we get that $\sigma({\rm FPdim}(X)) = {\rm FPdim}(X)$. Thus all simple objects in the orbit $(\sigma(X))$ has the same ${\rm FPdim}$, so the multiplicity $m$ of this basic ${\rm FPdim}$ satisfies $m \ge (p-1)/2$, i.e., $p \le 2m+1$. $\square$
Proof of Theorem 2:
According to [CGR] or [NN, Section 3], the rank $r$ of the Drinfeld center $\mathcal{Z}(\operatorname{Rep}(G))$ is determined by the number of irreducible characters within the centralizers of class representatives of $G$. This is equivalent to the count of conjugacy classes within the centralizers of these representatives and is also equal to the total number of conjugacy classes of pairs of commuting elements in $G$, as detailed in this post. Dave's comment shows that $r \ge \operatorname{ord}(g)$ for all elements $g$ in $G$ (because the pairs of commuting elements $(g,g^i)$ for $0 \le n < \operatorname{ord}(g)$ are all in distinct conjugacy classes), specifically implying that $r \ge p$ for all prime factors $p$ of $|G|$, by Cauchy's theorem. $\square$
Let me state the stronger version of Theorem 2 (following Goeff comment):
Theorem 2bis: Let $G$ be a finite group. Let $\Gamma_G$ be a complete set of conjugacy class representatives. Let $c_G$ be the number of conjugacy classes (i.e. $|\Gamma_G|$). Let $r_G$ be the rank of $\mathcal{Z}(\operatorname{Rep}(G))$ (so the number of conjugacy classes of pairs of commuting elements in $G$ by above). Let $Z(g)$ be the center of $G$. Then $$r_{G} \geq |Z(G)|c_G + \sum_{g \in \Gamma_{G} \backslash Z(G) } {\rm ord(g)}.$$Proof: The proposition in Appendix of this post states that $r_G = \sum_{a \in \Gamma_G} c_{C_G(a)}$, where $C_G(a)$ denotes the centralizer of $a$ in $G$. If $a \in Z(G)$ then $C_G(a) = G$ and so $c_{C_G(a)} =c_G $. Otherwise, in general we have $$c_{C_{G}(a)} \geq |Z(C_{G}(a))| \geq {\rm ord}(a).$$ The result follows. $\square$
Naive (former) proof of Theorem 1 (without using [DLN, Theorem II (iii)]):
As discussed in [NRWW, Section 3], a MTC $\mathcal{C}$ is associated with modular data $(S,T)$, which gives a projective representation of $${\rm SL}(2,\mathbb{Z}) = \langle s,t \ | \ (st)^3 = s^2, s^4 = e \rangle.$$ This representation can be lifted to a usual (linear) representation $\rho$ by utilizing the linear characters (i.e. one-dimensional representations), forming a cyclic group of order $12$. This representation is $r$-dimensional—where $r$ represents the rank of $\mathcal{C}$—and is congruence. This means it factors through ${\rm SL}(2,\mathbb{Z}/n\mathbb{Z})$, for some $n$ whose smallest one is called the level. The level is determined as ${\rm ord}(\rho(t))$, and as previously described, satisfies $${\rm ord}(T) \ | \ {\rm ord}(\rho(t)) \ | \ 12{\rm ord}(T).$$
A finite-dimensional congruence representation $\rho$ of level $n$ is completely reducible, hence it can be broken down into a direct sum of irreducible representations of ${\rm SL}(2,\mathbb{Z}/n\mathbb{Z})$. It's important to note that this includes only those irreducible representations that do not further factor through ${\rm SL}(2,\mathbb{Z}/d\mathbb{Z})$ for any proper divisor $d$ of $n$. Nevertheless, if $n = \prod_i p_i^{n_i}$ represents the prime factorization of $n$, then $\rho = \bigotimes_i \rho_i$ with each $\rho_i$ being a congruence representation of level $p_i^{n_i}$.
Although not necessary for this theorem, [NWW] proves that the finite-dimensional congruence representations are equivalent to symmetric ones, which are classified in [NWW2].
The dimensions $d$ of the irreducible finite-dimensional congruence representations at level $n = p^a$ are provided in the table at the end of [NW]. Observe that $d \ge (n-1)/2$ [with equality only if $a=1$], leading to $p \le p^a = n \le 2d+1$. Given that the rank $r$ of the MTC is the sum of dimensions $d$ of such irreducible representations, it follows that $d \le r$ and therefore $p \le 2r+1$.
According to the Cauchy theorem in [BNRW], the set $S$ of prime factors of ${\rm ord}(T)$ coincides with the prime factors of the norm $N$ of the categorical dimension of the MTC (of rank $r$). The prime numbers $p$ lastly mentioned (satisfying $p \le 2r+1$) constitute the set $S'$ of prime factors of the level $n$ of the congruence representation. Thus, $S \subseteq S' \subseteq S \cup \{2,3\}$, since ${\rm ord}(T) \ | \ n \ | \ 12{\rm ord}(T)$ and $12 = 2^2\cdot3$. Hence, for all prime factors $p \neq 2,3$ of $N$, it follows that $p \le 2r+1$. The inequality trivially holds for $p=2,3$. $\square$
References
[BNRW] Bruillard, P., Ng, S.-H., Rowell, E.C., Wang, Z.: Rank-finiteness for modular categories. J. Am. Math. Soc. 29(3), 857–881 (2016).
[CGR] A. Coste, T. Gannon, P. Ruelle. Finite group modular data. Nuclear Phys. B 581 (2000), no. 3, 679–717.
[DLN] C. Dong, X. Lin, S.H Ng, Congruence property in conformal field theory. Algebra Number Theory 9 (2015), no. 9, 2121--2166.
[NN] D. Naidu, D. Nikshych, Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups. Comm. Math. Phys. 279 (2008), no. 3, 845–872.
[NRWW] S.H. Ng, E.C. Rowell, Z. Wang, X.-G. Wen, Reconstruction of modular data from ${\rm SL}_2(\Bbb Z)$ representations. Comm. Math. Phys. 402 (2023), no. 3, 2465--2545.
[NWW] S.H. Ng, Y. Wang, S. Wilson. On symmetric representations of $\operatorname{SL}_2(\mathbb{Z})$. Proc. Amer. Math. Soc. 151 (2023), no. 4, 1415–1431.
[NWW2] Ng, S.-H., Wang, Y., Wilson, S.: SL2Reps, Constructing symmetric representations of SL(2, Z), Version1.0, Dec 2021. GAP package.
[NWZ] S.H. Ng, Y. Wang, Q. Zhang, Modular categories with transitive Galois actions. Comm. Math. Phys. 390 (2022), no. 3, 1271--1310.
[NW] A. Nobs, J. Wolfart, Die irreduziblen Darstellungen der Gruppen $\operatorname{SL}_{2}(\Bbb Z_{p})$, insbesondere $\operatorname{SL}_{2}(\Bbb Z_{2})$. II. (German) Comment. Math. Helv. 51 (1976), no. 4, 491–526.
[PSYZ] J. Plavnik, A. Schopieray, Z. Yu, Q. Zhang, Modular tensor categories, subcategories, and Galois orbits, arXiv:2111.05228.