p-adic Spin L-functions for GSp6
Ellen Eischen 1

http://www.elleneischen.com

February 26, 2021

Pair of workshops (in 2022) that might be of interest to people here: http://sites.google.com/view/automorphic2021/
1

This research was partially supported by NSF Grant DMS-1751281.
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Credits

This is joint work in progress with...

Giovanni Rosso

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Credits

This is joint work in progress with...

Giovanni Rosso

E. Eischen

Shrenik Shahp-adic Spin L-functions for GSp6

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Credits

This is joint work in progress with...

Giovanni Rosso

E. Eischen

Shrenik Shahp-adic Spin L-functions for GSp6

Willow Cat

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Also guest-starring and building on...
a variety of flavors of results by many different people...
especially an integral representation for the spin L-function on GSp6 forSiegel modular forms constructed by

Aaron Pollack

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Plan for this talk

Goal: Introduce a construction of p-adic Spin L-functions for GSp6 .
1Introduction to p-adic L-functions

2

Related results, especially for symplectic groups

3

Main results

4

Overview of key ingredients and steps of proofs of main results

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The Beginning: From Rational Numbers to Congruences
Euler (1700s): For 2k a positive even number▸−(2πi)⋅ B2k2k
ζ(2k) = 1 + 212k + 312k + 412k + ⋯ = 2⋅(2k−1)!

▸ζ(1 − 2k) = − B2k2k ∈ Q

2kKummer (1800s): For a prime p and 2k, 2k ′ positive even integers
such that (p − 1) ∤ 2k, 2k ′ ,2k ≡ 2k ′ mod φ(p d ) Ô⇒ ζ (p) (1 − 2k) ≡ ζ (p) (1 − 2k ′ ) mod p d ,where ζ (p) (1 − 2k) ∶= (1 − p 2k−1 )ζ(1 − 2k)

Kubota-Leopoldt (1960s): Construct p-adic zeta-function, in factp-adic Dirichlet L-functions
Serre (1970s): Construct p-adic zeta-functions, using modular forms
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Connection with Modular Forms
Eisenstein series of weight 2k ≥ 2:
G̃2k (z) ∶=12k(0,0)≠(m,n)∈Z×Z (mz + n)∑

Fourier expansion of G2k (z) ∶= (2πi)2k G̃2k (z):(2k−1)!

∞

G2k (z) = ζ(1 − 2k) + 2 ∑ σ2k−1 (n)q n ,n=1

where q = e 2πiz and σ2k−1 (n) = ∑d∣n d 2k−1 .
Special case of∞Gk,χ (q) = L(1 − k, χ) + 2 ∑ σk−1,χ (n)q n ,n=1

where χ is Dirichlet character and σ2k−1,χ (n) = ∑d∣n χ(d)d 2k−1 .
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Serre’s Innovation
Serre (1970s): Consider congruences between Fourier coefficients ofmodular forms (and construct p-adic modular forms...)

Theorem (Serre,1970s)
Suppose have modular forms
f (q) = a0 + a1 q + a2 q 2 + a3 q 3 + ⋯g (q) = b0 + b1 q + b2 q 2 + b3 q 3 + ⋯such that an ≡ bn mod p d for n ≥ 1. Then a0 ≡ b0 mod p d .
Apply to p-stabilization of Gk,χ :∞Gk,χ (q) ∶= L(p) (1 − k, χ) + 2 ∑ σk−1,χ (n)q n ,
(p)

(p)

n=1

where L(p) (1 − k, χ) ∶= (1 − χ(p)p k−1 )L(1 − k, χ) and(p)
σk−1,χ (n) ∶= ∑d∣n,p∤d χ(d)d k−1 . Recovers Kummer, Kubota-Leopoldt!
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The origins of the theory of p-adic modular forms
Theorem (Serre, 1973 Antwerp Volume 3)
Modular forms can be put into p-adic families, and you can take p-adic
limits of them. That is, if {fki (q) ∶= ∑n≥0 an (ki ) q n }i is a sequence ofmodular forms of weight ki with ki converging in Z/(p − 1)Z × Zp andlimi→∞ an (ki ) = an for n ≥ 1, then a0 (ki ) converge p-adically to some a0 .

RemarkBuilds on work of Hecke, Klingen, and Siegel on algebraicity and of Atkinand Swinnerton-Dyer on congruences.

RemarkThis is the first instance of properties of modular forms driving the studyof p-adic (or algebraic) properties of values of L-functions. (N.B. The
behavior of families of Galois representations is also tied to properties of
p-adic modular forms, as developed in work of Hida in the 1980s via
systems of Hecke eigenvalues.)
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More using modular forms to study L-functions
In analogue with how Klingen and Siegel’s work on algebraicity inspired
Serre’s approach to congruences and p-adic L-functions, we have:

Theorem (Shimura, 1970s)
Let f (q) = ∑n>0 an q n be a cusp form of weight k and g (q) = ∑n≥0 bn q n bea modular form of weight `. Consider the Dirichlet series∞D(s, f , g ) ∶= ∑ an bn n−s .
n=1

Then for ` < k and k+`−2<m<k2
D(m, f , g )∈Q
π k ⟨f , f ⟩Pet(in fact, lies in Q ({an , bn }n )).
Here ⟨, ⟩Pet denotes the usual Petersson pairing on modular forms.
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From algebraic to p-adic properties of Rankin–Selberg
convolutions
Theorem (Shimura, 1970s)
For each non-negative integer r such that ` + 2r < k,D(k − 1 − r , f , g ) = cπ ` ⟨f˜, g δλ Eλ ⟩Pet ,(r )where Eλ is a certain Eisenstein series of weight λ ∶= k − ` − 2r , δλ is a
Maass–Shimura differential operator, f˜ ∶= ∑n≥1 an q n , and c is aconstant dependent on k, `, r , and the level of f and g .
(r )Theorem (Hida, 1980s)
Reinterpret the Right Hand Side p-adically to construct p-adic L-functions(as p-adic measures).

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General philosophy

Connection between proofs of algebraicity and constructions of p-adic
L-functions

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Natural Questions, in View of Previous Examples
Question
For which (C-valued) L-functions L(s, M) and data M does there exist ap-adic function Lp−adic such that(∗)Lp−adic (n, M) = (∗′ )L(p) (n, M)
for all (n, M) meeting appropriate conditions (and such that theconstruction exploits congruences between Fourier expansions of modular
forms)?

Question
What can we say about algebraicity of values of L(n, M)? (We need toknow that values of algebraic, and furthermore p-integral, before talking
about p-adic properties.)

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Conjectures

Coates, Perrin-Riou:
Conjecture existence of wide class of p-adic L-functions
Realized as element of an Iwasawa algebra, i.e. a completed group ring(where group is profinite Galois group) over a p-adic ring
▸ Predicted form of the Euler factor at p▸▸

Greenberg: Building on the Main Conjecture of Iwasawa Theory(proved by Mazur and Wiles), predicted the meaning of many p-adic
L-functions through more extensive Main Conjectures, which concern
p-adic variation of arithmetic data
Deligne: Conjectures about algebraicity (again, necessary to establish
before talking about p-adic behavior)

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Some Answers, in the Form of Examples
1
2

Already saw example for Dirichlet L-functions (Kubota-Leopoldt)
There exists a p-adic L-function interpolating L(p) (s, χ) for
appropriate s and Hecke characters χ ∶ A×K → C× :
1
2
3

3

There exists a p-adic L-function interpolating L(p) (s, f , χ) for
appropriate s, χ, and modular forms f
1
2

4

Coates-Sinnott (1974): K real quadratic
Deligne-Ribet (1980): K totally real
Katz (1978): K CM, with splitting condition on the prime p

Hida (1980s, 1990s): f ordinary (in a Hida family)
Panchishkin (2000s): f finite slope (in a Coleman family)

There exists a p-adic L-function interpolating L(p) (s, π, χ) for
appropriate s, χ, and cuspidal automorphic representations π of aunitary group
1

Theorem of E–Harris–Li–Skinner (published in 2020): for π ordinary (ina Hida family), with splitting condition on the prime p

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In case this all sounds incremental...
Coates-Sinnot (case: real quadratic Hecke characters): 28 pages
Deligne-Ribet (case: totally real Hecke characters): 60 pages
Katz (case: CM Hecke character, with condition on p): 99 pages
Hida (case: ordinary modular forms): 36 pages...uses earlier work of Katz on p-adic families of Eisenstein series
Panchishkin (case: finite slope modular forms): 65 pages
All of the above papers were published in Inventiones.
E–Harris–Li–Skinner (case: ordinary cuspidal automorphic
representations of unitary groups): 160 page paper published inForum of Mathematics, Pi in 2020...uses about 100 pages of my earlier work (e.g. Crelle ’15, Algebra
Number Theory ’14) on p-adic families of Eisenstein series

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Strategy for constructing certain p-adic L-functions

This is the recipe employed by Hida, Panchishkin, E–Harris–Li–Skinner,and others:
1

Construct a p-adic family E of Eisenstein series (indexed by weights,
like the example of G2k at the beginning) on some group G

2

Restrict (“pull back”) to some (possibly smaller) subgroup H inside
G , and pair against a cusp form on H

3

Interpret (via an integral representation, such as the Rankin–Selberg
method) as a (familiar) L-function

This builds on a strategy for proving algebraicity results...

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Precursor to studying p-adic behavior: prove algebraicity
results!
Recipe for proving algebraicity results (pioneered by Shimura in his studyof Rankin–Selberg convolutions):
1

Realize your L-function in terms of an integral representation (e.g.
Rankin–Selberg convolution, doubling integral, ...), pairing an
Eisenstein series against cusp form(s), so values of your L-function are
realized as a pairing ⟨φ, E ⟩2

Find an Eisenstein series with algebraic Fourier coefficients

3

Find an othonormal basis {φi } for your space of cusp forms over Q̄,and show that ⟨φi , E ⟩/⟨φi , φi ⟩ ∈ Q̄. Use this prove similar statement
for any cusp form φ in place of φi .

It is nontrivial to carry out this recipe, and it is even nontrivial to acquire
some of the ingredients (in particular, the Fourier expansion of an
Eisenstein series, an integral representation of an L-function, andfurthermore an integral representation that is suitable for studying
algebraicity).
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What about symplectic groups GSp2n and Siegel modular
forms?
In analogue with the standard Langlands L-functions studied by
E–Harris–Li–Skinner in the setting of unitary groups, p-adic standard
Langlands L-functions associated to cuspidal automorphic forms on
symplectic groups were recently constructed by Zheng Liu, further
extending earlier work by Böcherer–Schmidt.
This relies on the doubling method, due to Piatetski-Shapiro–Rallisand earlier work of Garrett.
Shimura proved corresponding algebraicity results.

RemarkIn the setting of GSp2 = GL2 , the standard L-function is L(Sym2 f , s). On
the other hand, L(f , s) is a spin L-function....

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Spin L-functions for symplectic groups GSp2n

We have that L GSp2n = GSpin2n+1 , which has a natural 2n -dimensional
representation, the spin representation.

Question
What can we say about algebraicity and p-adic interpolation if we replace
L(s, π, std) (studied by Liu, Böcherer–Schmidt, ...) by L(s, π, spin), with
π a cuspidal automorphic representation of GSp2n ?
n = 1:This is L(f , s) with f a modular form. Using modular symbols,
Manin–Shimura proved algebraicity results, andMazur–Tate–Teitelbaum constructed p-adic L-function.
n = 2:Using higher Hida theory, together with Harris’s interpretationof L-values as cup products (useful for studying algebraicity),
Loeffler–Pilloni–Skinner–Zerbes constructed p-adic L-function.
n ≥ 3:algebraicity and p-adic properties for n = 3 is the subject of therest of this talk.Remain poorly understood for n > 3 (althoughBump–Ginzburg constructed an integral representation for n=3, 4, 5
that appears not to be amenable to proving algebraicity results)...
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p-adic spin L-functions for symplectic groups GSp6

This project focuses on analytic p-adic L-functions (obtained byp-adic interpolation).
On the algebraic side, there is recent related work of Cauchi–Jacinto
and Cauchi–Lemma–Jacinto.

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Main results
Theorem (E–Rosso–Shah)Let π be a cuspidal automorphic representation of GSp6 associated to aL(π,spin,s)Siegel modular form φ of weight 2r . Then π4s−6r +6 ⟨φ∣w ,φ⟩ is algebraic, forNs ∈ {3r − 2, . . . , 4r − 5}. (Here, wN is a certain involution for G .)Theorem (E–Rosso–Shah)
There is a two-variable p-adic L-function Lp (f(2r ) , j) that varies p-adicallyin 2r and j, interpolating the values
I∞ Ip

L(p) (π(2r ) , spin, s)⟨f(2r ) ∣wN , f(2r ) ⟩,where 3r − 2 ≤ j ≤ 4r − 5 and f(2r ) varies in a Hida family of ordinary Siegel
modular forms. Here I∞ denotes the archimedean Euler factor (a productof gamma factors) and Ip denotes the modified Euler factor at p.
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A Rankin–Selberg style pairing

Let φ be a cusp form associated to a holomorphic Siegel modular form on
GSp6 of positive, even scalar weight 2r . Consider the pairing∗⟨φ, E2r (g , s)⟩ = I2r(φ, s) = ∫GSp6 (Q)Z (A)/GSp6 (A)φ(g )E2r (g , s)dg ,for E2r (g , s) certain Eisenstein series on a group G (to be defined in a few
moments) restricted to GSp6 ⊂ G .

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Some things that are different in this setting

If you’re familiar with the doubling method for Siegel modular forms or the
Rankin–Selberg integral for modular forms, it might feel like you can use
what you know about those settings to proceed here.
There are some key differences, though:1

G is closely related to the exceptional group GE7 (and although G isrelated to a unitary group, that’s a quaternionic unitary group).

2

There’s currently no known moduli problem for G , no q-expansion
principle, etc.

3

The Rankin–Selberg style pairing here is a non-unique model.

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Preliminary setup
Fix:
a field F of characteristic 0
a quaternion algebra B over F
Define an F vector space
W ∶= F ⊕ H3 (B) ⊕ H3 (B) ⊕ F ,where H3 (B) denotes 3 × 3 hermitian matrices over B, i.e. matrices of theform∗⎛ c1 a3 a2 ⎞∗⎜a3 c2 a1 ⎟ ,⎝a2 a∗ c3 ⎠1with c1 , c2 , c3 ∈ F and a1 , a2 , a3 ∈ B. Here, a∗ denotes the conjugate of ain the quaternion algebra B.
The group G will be defined as a certain subgroup of GL(W ) × GL1 (F ).
First, though, a few conventions for operations on H3 (B)...
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Conventions for H3 (B)∗⎛ c1 a3 a2 ⎞Given h = ⎜a3∗ c2 a1 ⎟ ∈ H3 (B), we put⎝a2 a∗ c3 ⎠1N(h) ∶= c1 c2 c3 − c1 n(a1 ) − c2 n(a2 ) − c3 n(a3 ) + tr(a1 a2 a3 ),where n(a) ∶= aa∗ for all a ∈ B. We also define∗ ∗∗⎛c2 c3 − n(a1 ) a2 a1 − c3 a3 a3 a1 − c2 a2 ⎞h# ∶= ⎜ a1 a2 − c3 a3∗ c1 c2 − n(a2 ) a3∗ a2∗ − c1 a1 ⎟⎝ a∗ a∗ − c2 a2 a2 a3 − c1 a∗ c1 c2 − n(a3 )⎠1 3
1Given an element h′ ∈ H3 (B), we definetr(h, h′ ) ∶= trace(hh′ + h′ h).

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The group G
DefineA symplectic form ⟨, ⟩ ∶ W × W → F by⟨(a, b, c, d), (a′ , b ′ , c ′ , d ′ )⟩ ∶= ad ′ − tr(b, c ′ ) + tr(c, b ′ ) − da′ ,where tr(b, c) ∶= 12 (bc + cb) for all b, c ∈ H3 (B).
A quartic form Q ∶ W → F by
Q((a, b, c, d)) ∶= (ad − tr(b, c))2 + 4aN(c) + 4dN(b) − 4tr(b # , c # ).
We define G to be the similitude algebraic group consisting of all(g , ν(g )) ∈ GL(W ) × GL1 (F ) such that for all u, v , w ∈ W :⟨ug , vg ⟩ = ν(g )⟨u, v ⟩Q(wg ) = ν(g )2 Q(w )E. Eischenp-adic Spin L-functions for GSp6

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About the group G ...

There’s a group that is a double cover of G and GU6 (B)(quaternionic unitary group)
G is closely related to the exceptional group GE7
G is a half-spin group of type D6

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The Hermitian symmetric space for G (R)The Hermitian symmetric space for G (R) is
H ∶= {Z ∈ H3 (B ⊗R C)∣Z = X + iY , withX , Y ∈ H3 (BR ), Y positive definite }How does G act on H?
We have a map r ∶ H → W , Z ↦ (1, −Z , Z # , −N(Z )).
The action of G on H and the factor of automorphy jG (R) (g , Z ) aredefined simultaneously byr (Z )g −1 = jG (R) (g , Z )r (gZ ).
There is an embedding GSp6 ↪ G and corresponding embedding of Siegel
upper half space into H.

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Key input: Eisenstein series
1

We have an Eisenstein series E2r (g , s) that is holomorphic at s = r
and whose Fourier expansion when s = r has coefficients of the form
N(h)r −5

2

3

4

⎛
2r ⎞
∏ P` (` ) ,⎝`∣N(h)⎠with P` a polynomial with rational coefficients and h is a 3 × 3
hermitian quaternionic matrix.
We easily see that these Fourier coefficients satisfy
congruences...which would be nice for constructing an Eisensteinmeasure...BUT we don’t have a q-expansion principle here.
Pulls back nicely to the space for GSp6 , where we DO have aq-expansion principle and can construct a p-adic family of (pullbacksof) Eisenstein series
There is a differential operator D such that DE2r (g , s) = E2r +2 (g , s).
So it suffices to study D and E2r (g , r ). Has nice action in
q-expansions...
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Example: Calculating the action of the differential operator
We can show that the differential operator acting on a weight kmodular form f (Z ) on H has the formN(Y )−k ∆ (N(Y )k f ) ,where ∆ is the determinant of a matrix of partial derivatives.
Note the similarity with the familiar Maass–Shimura operators in thesetting of classical automorphic forms.
Given the way the action of G and the automorphy factor are defined,
how can we see that this operator raises the weight of a modular formon G by 2?
The answer follows from two key equations (whose proofs are a niceexercise in exploring how to think about the action of G and theembedding r ∶ H ↪ W ):For any α ∈ G + (R) and z, w ∈ H, we have:N(αz − αw ) = ν(α)−1 j(α, z)−1 j(α, w )−1 N(z − w )(1)

N(Im(αz)) = ∣j(α, Z )∣ ν(α) N(Imz)

(2)

−2

−1

(These rely on the observation that ⟨r (z), r (w )⟩ = N(z − w ).)E. Eischenp-adic Spin L-functions for GSp6

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Key realizations

Key realization that enables algebraicity theorem: Write⟨φ∣wN ,E2r (z,s+5−3r )⟩L(π,spin,s)with Eisenstein series E2r and a=⟨φ∣wN ,φ⟩π 4s−6r +6 ⟨φ∣wN ,φ⟩Rankin–Selberg style pairing as above, and recall that
DE2r (z, s) = E2r +2 (z, s) ...
Following Hida’s approach to constructing p-adic L-functions
associated to Rankin–Selberg convolutions of modular forms, we write⟨f ∣w ,E (z,s+5−3r )⟩, and we obtain a p-adic L-function`f2r (E2r (s)) = 2r N ⟨φ∣2rw ,f2r ⟩Nassociated to a Hida family of (ordinary) Siegel modular forms f(2r ) .

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Thank you!

For more information:
Visit http://www.elleneischen.com
Email eeischen@uoregon.edu
Also, check out the pair of workshops here:
http://sites.google.com/view/automorphic2021/

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About the image on the first slide

Figure: The Gaussian periods in the 255, 255-th cyclotomic extension of Q fixedby the action of the automorphism ζ ↦ ζ 254 , where ζ is a primitive 255, 255-throot of unity. Image credit: E. Eischen, based on earlier work by W. Duke, S. R.
Garcia, T. Hyde, and R. Lutz.

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